Page 125

"And so on, the symmetries and the reflections grow.

I lose myself in wonder at the beauty and the intricacy in which the numbers depend on each other, in secret. On the surface one would never imagine this closed circuit of intrigue. In the Mandala of nine, the digits have life-long partnerships, number ONE with EIGHT, TWO with SEVEN, THREE with SIX, FOUR with FIVE. Each pairing has an identical shape - when its multiplication sequence is plotted on the sigma circle, one is the reverse of the other.

Like partners in an eternal dance they turn, their bond being the number nine which holds them together and at the same time keeps them apart. Behind the whole revolution is this special mark, the character that keeps secrets and takes on several disguises - now a boundary, now a mirror, now unseen, and then taking complete hold of another's identity. Whichever way we turn in the circles. we find that all paths out of the labyrinth lead to (greek ommited) 9.

/ Page 125 / It is one grand act of conservation. Each orbit adds up to nine, as does each radiating arm. And in one magnificent homage, all the values of all the circles add up to nine. Everything in this hidden cosmos of numbers comes to the same thing, the uniqueness of 9.

Its component parts, the digits one to eight, hold in tight confidence

 Page 19

"At the age of 4, Pal Erdos remarked to his mother, 'If you subtract 250 from 100, you get 150 below zero.' Erdos could already multiply 3- and 4-digit numbers together in his head, but no one had taught him about negative numbers. 'It was an independent discovery,' he recalls happily..

Erdos grew up to be a great mathematician, but a surprising number of schoolchildren without his extraordinary talent will answer the question, 'How might this sequence continue: 876543210 .. .?' by suggesting, '1 less than nothing!' or 'minus 1, minus 2 . . .!'

Children in our society are floating in numbers. Whole numbers, fractions, decimals, approximations, estimations, record-breaking large numbers, minusculely small numbers. The Guinness Book of Records is a twentieth-century Book of Numbers, including the largest number in this Dictionary.

A mere handful of centuries ago numbers were smaller, fewer and simpler. It was seldom necessary to count beyond a few thousand. The Greek word myriad, which suggests a vast horde, was actually a mere 10,000, a fair size for an entire Greek army, but to us a poor attendance at a Saturday football match.

Fractions often stopped at one-twelfth. Merchants avoided finer divisions by dividing each measure into smaller measures, and the small measures into yet smaller, without going as far as Augustus de Morgan's fleas: 'Great fleas have little fleas upon their backs to bite 'em/And little fleas have lesser fleas, and so ad infinitum.'

The very conception of numbers proceeding to infinity, in any direction, appeared only in the imaginations of theologians and the greatest astronomers and mathematicians, such as Archimedes, who exhausted a circle with indefinitely many polygons and counted the grains of sand required to fill the universe.

To almost everyone else, numbers started at 1 and continued upwards in strictly one direction only, no further than ingenious systems of finger arithmetic, or the clerk's counting board, allowed.

(Zero, a strange and brilliant Indian invention, is not used for counting anyway. The Greeks had no conception of a zero number.)

These numbers were solid and substantial. To Pythagoras and his followers a number was always a number of things. To arrange a number such as 16 in a square pattern of dots was their idea of advanced and abstract mathematics.

John Tierney, 'Pal Erdos is in town. His brain is open', Science, October 1984.

/ Page 20 / To merchants also, numbers counted things.

To the later Greeks, numbers were still lengths of lines, areas of plane figures, or volumes of solids. What does a sphere with a volume -10 look like?

How could they make sense of numbers less than zero?

Early mathematicians did sometimes bump into negative numbers, in the dark as it were. They tried to avoid them, or pretended that they were not there, that they were an illusion.

Diophantus was a pioneer in number theory who still thought in strongly geometrical language. He solved many equations that to us have one negative and one positive root. He accepted the positive and rejected the negative. He 'knew' it was there, but it made no sense.

If an equation had no positive root, he rejected the equation. x + 10 = 5 was not a proper equation.

Perhaps it was a misfortune for a number-theorist to be born Greek. The Indians did not think of mathematics as geometry.

Hindu mathematicians first recognized negative roots, and the two square roots of a positive number, and multiplied positive and negative numbers together, though they were suspicious also.

Bhaskara commented on the negative root of a quadratic equation, 'The second value is in this case not to be taken, for it is inadequate; people do not approve of negative roots.'

On the other hand, the Chinese had already discovered negative numbers for counting purposes. By the twelfth century they were freely using red counting rods for positive quantities and black rods for negative, the exact opposite of our bank statements before computerization. They did not, however, recognize negative roots of equations.

As any schoolteacher will recognize, a chasm separates the simple act of counting backwards from the idea that negative numbers can be operated on in the same manner as positive numbers (with a couple of provisos).

How many generations of schoolchildren have never progressed further than the magic incantation, 'Two minuses make a plus!'

Craftsmen do not need negative numbers to measure backwards along a line. They turn their ruler round, or hold the ruler firmly and walk round the length they are measuring.

Merchants and bank clerks may easily juggle credits and debits without any conception that they are subtracting one negative number from another. Their intentions are honourably practical and concrete.

/ Page 21 / In fact, they made a practical contribution to the notation of mathematics. Our familiar plus and minus signs were first used in fifteenth- century German warehouses to show when a container was over or under the standard weight..

Number-theorists had a different problem. They met negative numbers stark naked, in the abstract. The number that when added to 10 makes 5 is just a number - or is it a fake number?

Renaissance mathematicians were as distrustful as Diophantus or Bhaskara.

Michael Stifel talked of numbers that are 'absurd' or 'fictitious below zero', which are obtained by subtracting ordinary numbers from zero. Descartes and Pascal agreed.

Yet, in the early Renaissance, one of the most difficult known problems was the solutions of equations, which often cried out for negative solutions. A few mathematicians accepted them, and even took a giant step further. Cardan was one.

The solutions to quadratic equations had been known since the Greeks, though Renaissance mathematicians continued to recognize three different types, illustrated by x²= 5x + 6; x² + 5x = 6, and x² + 6 = 5x. No negative coefficients!

The cubic equation was much harder.

Cardan, in his book The Great Art, still presented the cubic in more than a dozen different varieties, and solved them, using an idea he took from Tartaglia.

Yet he recognized negative numbers and even approached their square roots.

The very first square root of negative number on record, (square root) /81-144, is in the Stereometrica of Hero of Alexandria. Another, (square root) /1849 - 2016 was met by Diophantus as a possible root of a quadratic equation. They did not take them seriously. Neither did fifteenth-century European mathematicians.

Cardan proposed the problem: Divide 10 into two parts such that the product is 40.

He first said it was obviously impossible, but then solved it anyway, correctly giving the two solutions, 5 + (square root) / - 15 and 5 - (square root) / - 15.

He concluded by telling the reader that 'These quantities are "truly sophisticated" and that to continue working with them would be ''as subtle as it would be useless".'

The square roots of negative numbers! If negative numbers were false,

*Martin Gardner, 'Mathematical Games', Scientific American, June 1977.

/ Page 22 / absurd or fictitious, it is hardly to be wondered at that their square roots were described as 'imaginary'.

Even today, the theory of complex numbers is one of several hurdles that are recognized as separating 'elementary' from 'advanced' mathematics.

Pal Erdos's most famous proof is of the Prime Number theorem, which says that if n(x) is the number of primes not exceeding x, then as x tends to infinity,

tends to 1.

It was originally proved in 1896 using complex analysis. Here, 'complex' does not mean complicated, though it was, but using complex numbers. Erdos in 1949 published a proof that avoided complex numbers entirely. Such a proof is called 'elementary'. Here 'elementary' does not mean easy, merely that complex numbers are not used!

John Wallis accepted negative numbers but wrote of complex numbers, 'These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible.'

Wallis sounds (if I may say so) when talking of complex numbers (when he does) much like Bhaskara on numbers less than zero.

Mathematicians had reasons to be suspicious. Negative numbers, quintessentially - 1, do possess properties that positive numbers lack.

A friend of Pascal, Antoine Arnauld, argued that if negative numbers exist, then - 1/1 must equal 1/ - 1, which seems to assert that the ratio of a smaller to a larger quantity is equal to the ratio of the same larger quantity to the same smaller.

Most educated adults today would reject this idea after a moment's thought. No wonder this paradox was discussed at length.

Complex numbers are even more fiendish. Is (square root) / - 1 less than or greater than, say, 10? Neither, as Euler realized. The very idea of greater than or less than breaks down, and has to be reconstructed in a new form, a form incidentally that will also resolve Arnauld's paradox.

Fortunately, negative and complex numbers work, just as the calculator's red and black rods, or the warehouseman's + and - signs work.

Mathematicians were forced to accept negative and imaginary numbers, long before they had solved the conundrums that they posed.

/ Page 23 / Euler boldly used (square root) / - 1 in infinite series, and published his exquisite formula, eⁱⁿ = -1. He also introduced the letter i to stand for (square root) / - 1

Wessel, Argand and Gauss independently discovered around 1800 that complex numbers could be represented on a graph.

When Gauss introduced the term 'complex number' and expressed complex numbers as number pairs, their modem conception was almost complete.

F. Cajori, A History of Mathematical Notations, 2 vols., Open Court, 1977 (reprint); G. Cardan, Ars Magna (1545); and Augustus de Morgan, A Budget of Paradoxes (1872).

A mysterious number, which started life as a space on a counting board, turned into a written notice that a space was present, that is to say that something was absent, then confused medieval mathematicians who could not decide whether it was really a number or not, and achieved its highest status in modem abstract mathematics in which numbers are defined anyway only by their properties, and the properties of zero are at least as clear, and rather more substantial, than those of many other numbers.

The Babylonians in the second century BC used a system for mathematical and astronomical work in which the value of a numeral depended on its position. Two small wedges indicated that a place within a number was unoccupied, so distinguishing 207 from 27. (270 was distinguished from 27 by context alone.)

Whether this Babylonian system was transmitted to neighbouring cultures is not known.

Our system, in which the 0 is an extra numeral, originated in India. It was used from the second century BC to denote an empty place and as a numeral in a book by Bakhshali published in the third century.

The Sanskrit name for zero was sunya, meaning empty or blank, as it does today in some Indian languages. Translated by the Arabs as sifr, with the same meaning, it became the European name for nought, via the Latin zephirum, in different ways in different countries: zero, cifre, cifra, and the English words zero and cipher.

In AD 773 there appeared at the court of Caliph Al-Mansur in Baghdad an Indian who brought writings on astronomy by Brahma- gupta.

/ Page 24 / This was read by Al-Khwarizmi, the great Arab mathematician, whose name gave us the word 'algorithm' for an arithmetical process and more recently for a wider class of processes such as computers use, and who wrote a textbook of arithmetic in which he explained the new Indian numerals, published in AD 820.

At the other end of the Muslim world, in Spain at the beginning of the twelfth century, it was translated by Robert of Chester. This translation is the earliest known description of Indian numerals to the West.

There are several records of Arabic, that is, Indian, numerals being taught over the next century and a half. About 1240 they were even taught in a long and not very good poem. Yet they spread very slowly indeed, for two reasons.

The Arabic system did not just add a useful zero to the old Roman numerals; learners had to master the Arabic numerals 1 to 9 as well, and the zero numeral was a puzzle in itself.

Was zero a number? Was it a digit? If it stands for nothing, then surely it is nothing? But as every school pupil knows, if you add a harmless zero to the end of a number, you multiply it by 10! Our ten digits were often presented as the digits 1 to 9, plus the cypher, the zero: 'And there are nine figures that have value. . . and one more figure outside of them which is called null, 0, which has no value in itself but increases the value of others.'

The twelfth-century Salem Monastery manuscript had sounded a Platonic note: 'Every number arises from One, and this in turn from the Zero. In this lies a great and sacred mystery' though Plato started with One and knew nothing of any zero.

Merchants and bookkeepers had another reason to hesitate. To avoid tampering with written records, important amounts of money were written in full, in which case Indian numerals have no advantage, useful though they were for actual calculation.

A decisive step was taken by the first great mathematician of the Christian West, Leonardo of Pisa, called Fibonacci, who also features in this dictionary as the discoverer of the Fibonacci sequence.

Leonardo gives details of his life in his most famous book, the Liber Abaci. Leonardo's father was the chief magistrate of the Pisan trading colony at Bugia in Algeria. Leonardo spent several years in Africa, studying under a Muslim teacher. He also travelled widely to Greece, Egypt and the Middle East.

No doubt many merchants before Leonardo had noticed that the merchants they traded with used a very different system of numerals. / Page 25 / Leonardo compared the systems he met, and concluded that the Indian system he had learned in Africa was by far the best.

In 1202, and in a revised edition in 1228, he published his Book of Computation, the Liber Abaci, a compendium of almost all the mathematics then known.

In it he described the Indian system. Having learned of it as a merchant's son, he described its use in commercial arithmetic, in calculating proportions and mixtures, and in exchanging currency.

The final practical triumph of zero and its Indian numerals came with the spread of the printed book, and the rise of the merchant class.

Textbooks of arithmetic were among the most popular of the early printed books. They taught the merchant's children the skills with numbers that were becoming more and more essential at the same time as they gave the final push to counters and the counting board, and established the new numerals.

We so easily take zero for granted as a number, that it is surprising to consider that the Greeks had no conception of nothing, or emptiness, as a number, and doubly curious that this did not stop them, or many other cultures, from creating mathematics. Even when the Greeks treated limits and very small quantities, they had no conception of a quantity 'tending to zero'. It was sufficient that the quantity was less than another quantity, or might be made as small as desired.

Familiarity with zero did not exhaust its interest for mathematicians, who anyway had some problems in handling this extraordinary number.

Brahmagupta stated that 'positive or negative divided by cipher is a fraction with that for denominator'. This was called 'the quantity with zero as denominator'.

Mahavira wrote in his Compendium of Calculations: 'A number multiplied by zero is zero and that number remains unchanged which is divided by, added to or diminished by zero.' Did he think of division by zero as repeated subtraction, which had no effect?

The fact that zero added to or subtracted from a number left the number unchanged was a mystery directly comparable to the Py- thagoreans' refusal to accept 1 as a number, since it did not increase other numbers by multiplication.

Both these facts are part of the abstract definition of a field, of which ordinary numbers are an example. A field must contain a 'multiplicative identity', usually labelled 1 with the property that if g is any other element in the field, then 1 x g = g x 1 = g, and an 'additive identity', usually labelled 0, with the properties that for any g, 0 + g = g + 0 = g, and division by 0 is forbidden.

/ Page 25 /

0.110001000000000000000001000000000000000000

Like unity, 0 proves exceptional in other ways. It is an old puzzle to decide what 0° means. Since a° is always 1, when a is not zero, surely by continuity it should also equal 1 when a is zero?

Not so! 0^a is always 0, when a is not zero, so by the same argument from continuity, 0° should equal 0.

The values of functions such as 0! (factorial 0) are decided conventionally in order to make maximum sense and to be of maximum use.

The low status of zero in some circumstances is a great advantage to the lucky mathematician. When Lander and Parkin were looking for sums of 5 fifth powers whose sum was also a fifth power, one of their solutions included the number 0⁵. This solution immediately qualified, because powers of 0 do not count for obvious reasons, as a sum of 4 fifth powers equal to a 5th power, and destroyed a conjecture of Euler. (See 144.)

Karl Menninger. Number Words and Number Symbols, Massachusetts Institute of Technology Press, 1969.

/ Page 30 /

1

Unity

The Greeks did not consider 1, or unity, to be a number at all. It was the monad, the indivisible unit from which all other numbers arose. According to Euclid a number is an aggregate composed of units. Not unreasonably, they did not consider 1 to be an aggregate of itself.

As late as 1537, the German Kobel wrote in his book on computation, 'Wherefrom thou understandest that 1 is no number, but it is a generatrix, beginning, and foundation for all other numbers.'

The special significance of 1 is apparent in our language. The words 'one', 'an' and 'a' (a shortened form of 'an') are etymologically the same. So are the words 'unit', 'unity', 'union', 'unique' and 'universal', which all come from the Latin for one. It is no coincidence that these words are all exceptionally important in modern mathematics.

The Greeks considered that 1 was both odd and even, because when added to an even number it produced odd, and when added to an odd number it produced even. This reasoning is completely spurious, because any odd number has the same property. They were right, however, to notice that 1 is the only integer that produces more by addition than by multiplication, since multiplication by 1 does not change a number. In contrast, every other integer produces more by multiplication than by addition.

It is because multiplication by 1 does not change a number that 1 hardly ever appears as a coefficient in expressions such as x² + x + 4. It is pointless to write x as 1x, unless we wish to emphasize some pattern.

On the other hand, 1 is of vital significance when summing infinite series. The series,

1 + x + x² + x³ + x⁴ + x⁵ + x⁶ + . . .

has no sum if x is greater than 1, because each term is then greater than the previous term. If x = 1, then the series becomes

1+ 1 + 1 + 1 + 1 + 1 . . . and still has no sum. But when x is any number less than 1, then the sum of as many terms as we choose to add / Page 31 / approaches as closely as we wish to 1/(1 - x), without ever exceeding that number, and the infinite series has a finite sum.

What did the Greeks do about fractions? Surely they recognized that the indivisible unit, 1, could be divided into 2 parts, or 3 parts, or 59 parts? Not at all! They took the view that the original unit remained the same, while the result of the division, say 1/59, was taken as a new unit. Indeed, we still talk of a fraction whose numeractor is 1 as a unit fraction.

This interpretation fits the usage of merchants and craftsmen throughout the world. How much easier it is to consider 2 centimetres, rather than 0.02 metres, though they are mathematically the same! Psychologically, it is much simpler to invent new units of measure for small, and large, quantities, and completely avoid using very small or very large numbers.

1 appears in its modern disguise as the generatrix, the foundation of other numbers, in so many infinite sequences. It is, of course, the first square number, but it is also the first perfect cube, and the first 4th power, the first 5th power. . . the first of any power.

It is also the first triangular number, the first pentagonal number. . . the first Fibonacci number and the first Catalan number!

N. J. A. Sloane lists 2372 sequences which have been studied by mathematicians in his Handbook of Integer Sequences. With a minimum of fiddling he arranges for every sequence to start with the number 1.

Into how many pieces can a circular pancake be cut with n straight cuts? It is natural to start with the 1 piece, the whole pancake, which remains after zero cuts.

In how many ways can n objects be arranged in order? Modern mathematicians naturally start with 1 object, which can be 'arranged' in just 1 way. The Greeks would undoubtedly have argued, very plausibly, that the sequence should start with 2 objects, which can be arranged in order in 2 ways. They would have claimed that 1 object cannot be arranged in any order at all.

1 is especially important because of its lack of factors. This suggests that it should be counted as a prime number, because it fits the definition, 'A prime number is divisible by no number except itself and 1', but once again 1 is usually considered to be an exception.

A conventional reason depends on an important and favourite theorem, that any number can be written as the product of prime factors in only one way, apart from different ways of ordering the factors. Thus 12 = 2 x 2 x 3 and no other product of prime numbers equals 12.

This theorem would have to be adjusted if 1 were a prime, because / Page 32 / then 12 would also equal 1x 2 x 2 x 3, and 1 x 1 x 2 x 2 x 3 and so on. Untidy! So 1 is dismissed from the list of primes.

Euler had a different reason for rejecting 1. He observed that the sum of the divisors of a prime number, p, is always p + 1, the prime p itself and the number 1. The exception, of course, to this rule turns out to be 1. The simplest way to dispose of this exceptional case is to deny that 1 is prime.

Because 1 is so small, as it were, and has no factors apart from itself, it does not feature in many of the properties in this dictionary. To write 1 as the sum of two squares, it is necessary to write 1 = 1² + 0² which is trivial. In the same way, 1 can be written as the sum of 3 squares, or even of 5 cubes, which is even more boring.

Similarly, 1 is the smallest number that is simultaneously triangular and pentagonal. Also boring!

Indeed, 1 might be considered to be the first number that is both boring and interesting.

Yet it does appear in this dictionary in a small but essential way.

Precisely because it has no factors, it is never obvious whether expressions such as 2⁵ - 1, the 5th Mersenne number, or 2²³ + 1, the 3rd Fermat number, will have any factors.

When Euclid wanted to show that the number of primes is unlimited, he considered three primes, by way of example. Call them, A, Band C. Multiply them together, and add 1: is ABC + 1 prime? If so there is a prime larger than any of A, B or C. If ABC + 1 is not prime, then it has a prime factor, which cannot be any of the primes A, B or C. So there is at least one more prime. . .

Euclid's argument would not have worked if he had considered ABC + 2, or ABC + 3. Only 1 will guarantee his argument.

Our number line, familiar to children in school, extends at least from 0 to infinity, and the gaps between the whole numbers are filled by infinities of fractions, irrational numbers, and even more transcendental numbers.

The Greeks' idea of number was simpler and inadequate for the purposes of modern mathematicians. Yet one great mathematician saw the whole numbers, starting with 1, as the only real numbers. 'God made the integers,' claimed the nineteenth-century mathematician Kronecker. 'All the rest are the work of man.'

1 is not the first number in this dictionary, but in its own way it is the foundation on which all the other entries are based.

Karl Menninger, Number Words and Number Symbols. Massachusetts Institute of Technology Press, 1969.

/ Page 41 /

2

The number 2 has been exceptional from the earliest times, in many aspects of human life, not just mathematically.

It is distinguished in many languages, for example in original Indo- European, Egyptian, Arabic, Hebrew, Sanskrit and Greek, by the presence of dual cases for nouns, used when referring to 2 of the object, rather than 1or many. A few languages also had trial and quatemal forms.

The word two, when used as an adjective, was often inflected, as were occasionally the words three and four.

Modem languages reflect the significance of 2 in words such as dual, duel, couple, pair, twin and double.

The early Greeks were uncertain as to whether 2 was a number at all, observing that it has, as it were, a beginning and an end but no middle.

More mathematically, they pointed out that 2 + 2 = 2 x 2, or indeed that any number multiplied by 2 is equal to the same number added to itself. Since they expected multiplication to do more than mere addition, they considered 2 an exceptional case.

Whether 2 qualified as a proper number or not, it was considered to be female, as were all even numbers, in contrast to odd numbers, which were male.

Division into two parts, dichotomy, is more significant psychologically and more frequent in practice than any other classification.

The commonest symmetry is bilateral, two-sided about a single axis, and is of order 2.

Our bodies are bilaterally symmetrical, and we naturally distinguish right from left, up from down, in front from behind. Night is separated from day, there are two sexes, the seasons are expressed in pairs of pairs, summer and winter separated by spring and autumn, and comparisons / Page 42 / are most commonly dichotomous, such as stronger or weaker than, better or worse than, youth versus age and so on.

2 and division into 2 parts is just as significant in mathematics. 2 is the first even number, all numbers being divided into odd and even.

The basic operations of addition, subtraction, multiplication and division are binary operations, performed in the first instance on 2 numbers.

By subtraction from zero, every positive number is associated with a unique negative number, and 0 divides all numbers into positive and negative. Similarly division into 1 associates each number with its reciprocal.

2 is the first prime and the only even prime.

2 is a factor of 10, the base of the usual number system. Therefore a number is divisible by 2 if its unit digit is, and by 2ⁿ if 2ⁿ divides the number formed by its last n digits.

Powers of 2 appear more frequently in mathematics than those of any other number.

An integer is the sum of a sequence of consecutive integers if and only if it is not a power of 2.

The first deficient number. All powers of a prime are deficient, but powers of 2 are only just so.

Euler asserted what Descartes had supposed, that in all simple poly- hedra,- for example the cube and the square pyramid, the number of vertices plus the number of faces exceeds the number of edges by 2.

Fermat's last theorem states that the equation xⁿ + yⁿ = zⁿ has solutions in integers only when n = 2. The solutions are then sides of a right-angled Pythagorean triangle.

Fermat's equation being exceedingly difficult to solve, several mathematicians have noticed in an idle moment that n^x + n^y = n^z is much easier. Its only solutions in integers are when n = 2, and 2¹ + 2¹ = 2².

Goldbach conjectured that every even number greater than 2 is the sum of 2 prime numbers.

The binary system

The English imperial system of measures used to contain a long sequence of measures, some of which are still in use, in which each measure was double the previous one. Presumably they were very useful in practice, though it is unlikely that most merchants had any idea how many gills were contained in a tun:

1 tun = 2 pipes = 4 hogsheads = 8 barrels = 16 kilderkins = 32 / Page 43 / firkins or bushels = 64 demi-bushels = 128 pecks = 256 gallons = 512 pottles = 1024 quarts = 2048 pints = 4096 chopins = 8192 gills.*

The numbers appearing in this list are just powers of 2, from 2⁰ = 1 up to 2¹³ = 8192.

These measures could very easily have been expressed in binary notation, or base 2.

Every number can be expressed in a unique way as the sum of powers of 2. Thus: 87 = 64 + 16 + 4 + 2 + 1, which can be written briefly as 87 = 1010111.

Each unit indicates a power of 2 that must be included and each zero a power that must be left out, as in this chart for 1010111:

The binary system was invented in Europe by Leibniz, although it is referred to in a Chinese book which supposedly dates from about.3000 BC,

Leibniz associated the1 with God and the 0 with nothingness, and found a mystical significance in the fact that all numbers could thus be created out of unity and nothingness. Without accepting his mathematical theology we can appreciate that there is immense elegance and simplicity in the binary system.

As long ago as 1725 Basile Bouchon invented a device that used a roll of perforated paper to control the warp threads on a mechanical loom. Any position on a piece of paper can be thought of as either punched or not-punched. The same idea was used in the pianola, a mechanical piano popular in Victorian homes, which was also controlled by rolls of paper.

The looms were soon changed to control by punched cards, which were also used in Charles Babbage's Analytical Engine, a forerunner of the modern digital computer, which relied on punched cards until the arrival of magnetic tapes and discs. Binary notation is especially useful in computers because they are most simply built out of components that have two states: either they are on or off, full or empty, occupied or unoccupied,

The same principle makes binary notation ideal for coding messages to be sent along a wire. The1 and 0 are represented by the current being switched on and off.

Long before mechanical computers were invented, the Egyptians

*Keith Devlin, Guardian, 20 October 1983.

/ Page 44 / multiplied by doubling, as many times as necessary, and adding the results. For example, to multiply by 6 it is sufficient to double twice, and add the two answers together. Within living memory, Russian peasants used a more sophisticated version of the same idea, which was once used in many parts of Europe.

To multiply 27 by 35, write the numbers at the top of two columns: choose one column and halve the number again and again, ignoring any remainders, until 1 is reached. Now double the other number as many times:

Cross out the numbers in this second column that are opposite an even number in the first. The sum of the remaining numbers is the answer, 945.

Page 46

3

The first odd number according to the Greeks, who did not consider unity to be a number.

To the Pythagoreans, the first number because, unlike 1 and 2, it possesses a beginning, and middle and an end. They also considered 3, and all odd numbers, to be male, in contrast to even numbers, which were female.

The first number, according to Proclus, because it is increased more by multiplication than by addition, meaning that 3 x 3 is greater than 3 + 3.

Division or classification into 3 parts is exceptionally common. In many languages, the positive, comparative and superlative are dif- ferentiated. In English the sequence once-twice-thrice goes no further.

There were trinities of gods in Greece, Egypt and Babylon. In Christianity, God is a trinity.

In Greek mythology there were 3 Fates, 3 Furies, 3 Graces, 3 times 3 Muses, and Paris had to choose between 3 goddesses.

Oaths are traditionally repeated 3 times. In the New Testament, Peter / Page 47 / denies Christ three times. The Bellman in 'The Hunting of the snark' says, more prosaically, 'What I tell you three times is true!'

The world is traditionally divided into three parts, the underworld, the earth, and the heavens.

The natural world is 3 dimensional, Einstein's 4th dimension of time being unsymmetrically related to the 3 dimensions of length. In 3 dimensions, at most 3 lines can be drawn that are mutually perpendicular.

The Greeks considered lengths, the squares of lengths, which were represented by areas, and the cubes of lengths, represented by solids. Higher powers were rejected as unnatural. Numbers with 3 factors were sometimes considered as solid, just as a number with 2 factors was interpreted by a plane figure, such as a square or some shape of rectangle, or by one of the polygonal figures.

(A commentator on Plato describes even numbers as isosceles, because they can be divided into equal parts, and odd numbers as scalene.)

They also associated 3 with the triangle, which has 3 vertices and 3 edges, and was the commonest figure in their geometry and ours.

The trisection of the angle was one of the three famous problems of antiquity, the others being the squaring of the circle, and the duplication of the cube.

The problem is, or was, to trisect an arbitrary angle, using only a ruler, meaning an unmarked straight edge, and a pair of compasses. Like the duplication of the cube, it depends, in modern language, on the solution of a cubic equation.

Descartes showed that this can be accomplished as the intersection of a parabola and a circle, but unfortunately the required points on the parabola cannot be constructed by ruler and compasses.

It can however be solved by the use of special curves. Pappus used a hyperbola, and Hippias invented the quadratrix which can be used to divide an angle in any proportion. The conchoid invented by Nicomedes will trisect the angle and duplicate the cube.

Euler proved that in any triangle, the centroid lies on the line joining the circumcentre to the point of intersection of the altitudes, and divides it in the ratio 1: 2.

A circle can be drawn through any 3 points not on a straight line.

There are just 3 tesselations of the plane with regular polygons, using equilateral triangles, squares, or hexagons as in a honeycomb.

3 is the second triangular number, after the inevitable 1. Gauss proved that every integer is the sum of at most 3 triangular numbers. The 18th

/ Page 55 /

4

The first composite number, the second square, and the first square of a prime..

The Pythagoreans called numbers divisible by 4, even-even. For this / Page 56 / reason, 4, and also 8, were associated with harmony and justice, in contrast to the scales that symbolize justice in modern Western law.

4 is also associated by the Pythagoreans with the tetraktys, the pattern of the first 4 numbers arranged in a triangle.

They postulated 4 elements, earth, air, fire and water, symbolized respectively by the cube, octahedron, tetrahedron and icosahedron. The remaining Platonic solid, the dodecahedron, was associated with the sphere of the fixed stars, and later with the quintessence of the medieval alchemists.

A person's temperament was determined by combinations of 4 humours.

Being 2 by 2, there are 4 cardinal points of the compass and 4 corners of the world, and 4 winds.

In the Old Testament there were 4 rivers of paradise, one for each direction, supposed to prefigure the 4 gospels of the New Testament.

The quadrivium of Plato divided mathematics, in his general sense of higher knowledge, into the discrete and the continuous. The absolute discrete was arithmetic, the relative discrete was music. The stable continuous was geometry and the moving continuous, astronomy.

The most pleasing musical intervals are associated with the ratios of the numbers 1 to 4.

The Greeks also associated 4 with solid objects, notwithstanding their association between 3 and volume. They followed the natural progression, 1 for a point, 2 for a line, 3 for a surface, and 4 for a solid.

The simplest Platonic solid, the tetrahedron, has 4 vertices and 4 faces. A square has 4 edges and 4 vertices. A cube has square faces, while its dual, the octahedron, has 4 faces about each vertex.

Being 2², a plane figure with bilateral symmetry about two different -lines is divided into 4 congruent parts.

Einstein's space-time is 4-dimensional. However, in recent theories, 4 dimensions are insufficient.

A hyperbola can be drawn through any 4 points in the plane, no three of which are colinear.

Every integer is the sum of at most 4 squares. This celebrated theorem may have been known empirically to Diophantus. Bachet tested it successfully up to 120 and stated it in his edition of Diophantus, to which he added some of his own material.

It was studied by Fermat and Euler, who failed to solve it, and finally proved by Lagrange in 1770.

Only one-sixth of all numbers, those of the form 4ⁿ(8m + 7), however, / Page 57 / actually require 4 squares. The remainder are the sum of at most 3 squares.

Ferrari first solved equations of the 4th degree. His solution was published by Cardan in his Ars Magna.

The general equation of higher degree cannot be solved by the use of radicals.

The 4-colour problem

For more than a century the 4-colour conjecture was one of the great unsolved problems of mathematics. Some mathematicians would still say that it has not been solved satisfactorily.

In October 1852, Francis Guthrie was colouring a map of England. It suddenly occurred to him to wonder how many colours were needed if, as is natural, no two adjacent counties were given the same colour. He supposed the answer was 4.

It was published in 1878, setting in motion a bizarre but not untypical sequence of events.

Kempe thought that he had proved it in 1879, but eleven years later his proof was shown to be faulty. Meanwhile, in 1880, the conjecture had been proved again, but this proof was also flawed.

However, these attempts were valuable in deepening mathematicians' understanding of the problem. Indeed, many important concepts in graph theory were developed through attacks on this problem, which however proved extremely resistant.

The solution was finally achieved in 1976 by Wolfgang Haken and Kenneth Appel who transformed the problem into a set of sub-problems that could be checked by computer.

Mathematicians have been sceptical because of the lengthy mathematical reasoning involved, and the length of time, 1200 hours, taken on the computer. The very existence of a proof that few other mathematicians will ever be able to check is a recent development in mathematics. Another example of the same phenomenon is the classification of finite groups. This classification is now complete but the entire proof is spread across thousands of pages in different journals published over the years. This contradicts the traditional idea of a proof as an available means of confirming a thesis and persuading others also that it is true.

4 is exceptional in not dividing (4 - 1)! = 3!. It is the only composite n which does not divide (n - 1)!.

Brocard's problem asks: When is n! + 1 a square? 4! + 1 = 5².

/ Page 58 /

4-123 105. . .

A number is divisible by 4 if the number represented by its last two digits is divisible by 4.

Starting with any number, form a new number by adding the squares of its digits. Repeat.

This process eventually either sticks on 1, or goes round a loop of which 4 is the smallest member: 4 - 16 - 37 - 58 - 89 - 145 - 42 - 20 - 4.. .

If a number in base 10 is a multiple of its reversal, their ratio is either 4 or 9.

4 is the only number equal to the number of letters in its normal English expression: 'four'.

4-123105 . . .

(square root) / 17, the highest root to be proved irrational by Theodorus.

5

The Pythagoreans associated the number 5 with marriage, because it is the sum of what were to them the first even, female number, 2, and the first odd, male number, 3.

5 is the hypotenuse of the smallest Pythagorean triangle, that is, a right-angled triangle with integral sides.

The Pythagoreans also associated this triangle with marriage and Pythagoras' theorem was sometimes called the Theorem of the Bride. The sides 3 and 4 were associated with the male and female respectively, and the hypotenuse, 5, with the offspring.

The 3-4-5 triangle is the only Pythagorean triangle whose sides are in arithmetical progression, and the only one whose area is one-half of its perimeter.

The mystic pentagram, which was so important to the Pythagoreans, was known in Babylonia and probably imported from there.

The Pentagram was associated with the division of line in extreme and mean proportion, the Golden Section, and also with the fourth of the regular solids, the dodecahedron, whose faces are regular pentagons. The early Pythagoreans did not know the fifth regular Platonic solid, the icosahedron.

By constructing a nest of pentagrams inside a regular pentagon, it is relatively easy to show that subtraction of the sides and diagonals can be continued indefinitely. It has been suggested that this pattern led to the idea that some lengths are incommensurable.

The Pythagoreans, according to Plutarch, also called 5 nature, because / Page 59 / when multiplied by itself, it terminates in itself. That is, all powers of 5 end in the digit 5. They knew that 6 shares this property, but no other digit.

In modern terminology, 5 and 6 are the smallest automorphic numbers.

5 is the sum of two squares, 5 = 1² + 2², like any hypotenuse of a Pythagorean triangle.

It is also a prime, the first, of the form 4n + 1, from which it follows that it is the sum of two squares in one way only.

5 is the first prime of the form 6n - 1. All primes are one more or one less than a multiple of 6, except 2 and 3.

Pappus showed how to construct a conic through any 5 points in the plane, no 3 of which are colinear.

5 is the second Fermat number and the second Fermat prime:

5 = 2² + 1. Only 5 Fermat primes are known to exist.

The 5th Mersenne number, 2⁵ - 1 = 31 and is prime, the third to be so, leading to the third perfect number, 496.

5! + 1 is a square.

Every number is the sum of 5 positive or negative cubes in an infinite number of ways.

The general algebraic equation of the 5th degree cannot be solved in radicals. First proved by Abel in 1824.

Lame showed that the Euclidean algorithm for finding the highest common factor of two numbers takes in base 10 at most 5 times as many steps as there are digits in the smallest number.

5 is a member of two pairs of twin primes, 3 and 5, and 5 and 7.

/ Page 60 / 5-11-17-23 is the smallest sequence of 4 primes in arithmetical progression. Add the prime 29 to form the smallest set of 5 primes in arithmetical progression.

5 is probably the only odd untouchable number.

The volume of the unit 'sphere' in hyperspace increases up to 5-dimensional space, and decreases thereafter.

Counting in 5s

This might seem a natural base for a counting system, since we have 5 fingers per hand. However, only one language uses a counting system based exclusively on 5, Saraveca, a South American Arawakan language, though 5 has a special significance in many counting systems based on 10 and 20. For example in many Central American languages, the numbers 6 through 9 are expressed as 5 + 1, 5 + 2 and so on.

The Romans used V = 5, L = 50 and D = 500, so 664 was DCLXIIII. (The idea of placing an I before V to represent 4, or I before X for 9, for example, which makes numbers shorter to write while making them more confusing for arithmetic, was hardly ever used by the Romans themselves and became popular in Europe only after the invention of printing.)

Divisibility

Because 5, like 2, is a factor of 10, decimal fractions such as 1/20, whose denominators are products of 2s and 5s only, have finite decimal expansions and do not recur.

More precisely, if n = 2^p5^q, then the length of 1/n as a decimal is the greater of p and q.

If 1/m is a recurring decimal, and 1/n terminates, then l/mn has a nonperiodic part whose length is that of 1/n, and a recurring part whose length is the period of 1/m.

The Platonic solids

There are 5 Platonic solids, the regular tetrahedron, cube, octahedron, dodecahedron and icosahedron, all but the cube being named after the Greek word for their number of faces.

They were all known to the Greeks. Theaetetus, a pupil of Plato, showed how to inscribe the last two in a sphere. Euclid showed, by considering the possible arrangements of regular polygons around a point, that there are no more than 5.

Kepler used them, with typical confidence in their mystical properties, to explain the relative sizes of the orbits of the planets:

/ Page 61 / The earth's orbit is the measure of all things; circumscribe around it a dodeca- hedron, and the circle containing this will be Mars: circumscribe around Mars a tetrahedron, and the circle containing this will be Jupiter: circumscribe around Jupiter a cube, and the circle containing this will be Saturn. Now inscribe within the earth an icosahedron, and the circle contained in it will be Venus; inscribe within Venus an octahedron, and the circle contained in it will be Mercury. You now have the reason for the number of planets.

The idea of a polyhedron can be extended to more than 3 dimensions, just as a polyhedron can be considered as a 3-dimensional polygon.

There are 5 cells in the simplest regular 4-dimensional polytopes, called the simplex, which also has 10 faces, 10 edges and 5 vertices, so that it is self-dual.

The Fibonacci sequence

5 is the fifth Fibonacci number.

Leonardo of Pisa, called Fibonacci, discussed in his Liber Abaci this problem:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Assuming that the rabbits are immortal, the number at the end of each month follows this sequence. (Leonardo omitted the first term, supposing that the first pair bred immediately.)

1 1 2 3 5 8 13 21 34 55 89 144 233 ...

It was christened the Fibonacci sequence by Eduard Lucas in 1877, when he used it, and another sequence now named after himself, to search for primes among the Mersenne numbers.

It is one of the curious coincidences that occur in the history of mathematics that a problem about rabbits should generate a sequence of numbers of such interest and fascination. Rabbits, needless to say, do not feature again in its history.

Its first and simplest property is that each term is the sum of the two previous terms. Thus the next term will be 144 + 233 = 377. This was surely known to Fibonacci, though he nowhere states it. Mathematicians do not always state the obvious.

Kepler believed that almost all trees and bushes have flowers with five petals and consequently fruits with five compartments. He naturally associated this fact with the regular pentagon and the Divine Proportion. He continues;

/ Page 62 / It is so arranged that the two lesser terms of a progressive series added together constitute the third. . . and so on to infinity, as the same proportion continues unbroken. It is impossible to provide a perfect example in round numbers. However. . . Let the smallest numbers be 1 and 1, which you must imagine as unequal. Add them, and the sum will be 2: add to this 1, result 3; add 2 to this, and get 5; add 3, get 8. . . As 5 is to 8, so 8 is to 13, approximately, and as 8 is to 13, so 13 is to 21, approximately.

This statement could scarcely be clearer, but it was not until 1753 that the Scottish mathematician Robert Simson first stated explicitly that the ratios of consecutive terms tend to a limit, which is (symbol ommitted), the Golden Ratio. These are the first few ratios: 1/1 2/1 3/2 5/3 8/5 13/8 21/13 34/21 55/34 89/55 144/89 233/144 ...

Successive ratios are alternately less than and greater than the Golden Ratio. After 12 terms the match with (symbol ommitted) is correct to 4 decimal places. For much higher values the Fibonacci sequence matches the geometric sequence (symbol ommitted) very closely indeed.

(This is a consequence only of the rule that each term is the sum of the two preceding terms. Start with any two numbers, construct a generalized Fibonacci series, by adding successive terms to get the next, and their ratio will tend to (symbol ommitted) )

/ Page 63 / The number Fibonacci identifies is literally endless.

Lucas discovered a relationship between Fibonacci numbers and the binomial coefficients:

/ Page 64 / Charles Raine ingeniously connected Fibonacci numbers to Pythagorean triangles. Take any 4 consecutive Fibonacci numbers; the product of the outer terms and twice the product of the inner terms are the legs of a Pythagorean triangle: for example, 3, 5, 8, 13, gives the two legs, 39 and 80, of the right-angled triangle 39-80-89. The hypotenuse, 89, is also a Fibonacci number! Its subscript is half the sum of the subscripts of the four original numbers. Finally, the area of the triangle is the product of the original four numbers, 1560.

/ Page 65/ The Fibonacci sequence is also linked in a surprising way with the growth of plants. Kepler may have realized this. He writes:

It is in the likeness of this self-developing series that the faculty of propagation is in my opinion, formed; and so in a flower the authentic flag of this faculty is shown, the pentagon. I pass over all the other arguments that a delightful rumination could adduce in proof of this.

What were Kepler's other arguments? He does not say, but in the nineteenth century Schimper and Braun investigated phyllotaxis, the arrangements of leaves round a stem.

Leaves grow in a spiral, such that the angles between each pair of successive leaves are constant. The commonest angles are 180°, 120°, 144°,135°,138°27',137°8', 137°38', 137°27', 137°31'... which seem to be tending to a limit.

/ Page 66 / What that limit is becomes clearer when they are expressed as ratios of a complete circle.

These ratios are, respectively, 1/2, 1/3, 2/5, 3/8, 5/13, 8/21, 13/34, 21/55 and 34/89, the ratios of alternate members of the Fibonacci series.

To put that another way, the numerator and denominator of each new fraction are sums of the numerators and denominators of the previous two fractions. These ratios tend to the limiting value (symbol ommitted) -2, and the limiting angle is approximately 137°30' and 28 seconds, which divides the circumference of a circle in the Golden Ratio.

The smallest ratios, 1/2 and 1/3, are found in grasses and sedges but are otherwise not very common, though commoner than 1/4 and 1/5, which do exist and form part of another Fibonacci-type sequence.

The most frequent leaf arrangements are 2/5, found in roses, and 3/8. Much higher ratios, however, appear much more clearly in the scales of a fir cone or the florets of a sunflower, which are packed closely together. The packing is highly regular, forming sets of spiral rows, or parastichies, two of which are more prominent than the rest.

A pineapple usually has 8 and 13 parastichies. A sunflower may have from 21/34 up to as high as 89/144. Even 144/233 has been claimed for one giant plant.

(Although plants of the same species and even of the same family tend to have the same parastichy numbers, the higher numbers especially do vary from plant to plant. The phyllotaxis may even change as a plant grows, starting with a low ratio such as 1/2 or 1/3 and then changing to higher ratios.)

Why do plants grow this way? Less entranced by the Fibonacci numbers than mathematicians, botanists are more interested in an explanation, on which they do not yet agree.

One plausible theory, which might be explained by chemical inhibition of growth, is that each primordium, the primitive leaf bud, develops in the largest gap available. Whatever the botanists eventually decide, mathematicians will continue to delight in this connection between rabbits and the plants they eat.

The Fibonacci numbers have other uses in more advanced mathematics.

The Russian Matasyevic used Fibonacci to finally solve Hilbert's 10th problem. No algorithm exists that, given any Diophantine equation, will decide within a finite number of steps whether it has a solution. He exploited the rate at which the sequence of Fibonacci numbers increases.

/ Page 67 / They have also recently found further uses in computer science, in designing efficient algorithms for constructing and searching tables of data, for example.

Johannes Kepler, The Six-cornered Snowflake, Oxford University Press, Oxford, 1966.

5.256 946 404 860...

The approximate 'volumes' of the unit 'spheres' in dimensions from 1 upwards are:

The volume is a maximum in 5 dimensions, and declines thereafter.

If however the dimension is regarded as a real variable, able to take non-integral values, then the maximum volume occurs in 'space' of this dimension, 5.256. . .

The volume is then 5.277768 ... compared to the volume in 5 dimensions of 5.263789 . . . [David Singmaster]

6

The second composite number and the first with 2 distinct factors.

Therefore the first number, apart from 1, which is not the power of a prime.

The Pythagoreans associated 6 with marriage and health, because it is the product of their first even and first odd numbers, which were female and male respectively.

It also stood for equilibrium, symbolized by two triangles, base to base.

It is the area and the semi-perimeter of the first Pythagorean triangle, with sides, 3, 4, 5.

The first perfect number, as defined by Euclid. Its factors are 1, 2, 3 and 6 = 1 + 2 + 3.

It is the only perfect number that is not the sum of successive cubes.

St Augustine wrote, 'Six is a number perfect in itself. . . God created all things in six days because this number is perfect. And it would remain even if the work of six days did not exist.' [Bieler]

6 is also equal to 1 x 2 x 3, and is therefore the 3rd factorial, 3!, and also the second primorial.

No other number is the product of 3 numbers and the sum of the same 3 numbers.

/ Page 68 / 1,2, 3 is also the only set of 3 integers such that each divides the sum of the other two.

6 also equals (square root) / (1³ + 2³ + 3³).

It is the only number that is the sum of exactly 3 of its factors, which is the same as saying that 1 can be expressed uniquely as the sum of 3 unit fractions, the smallest of which is 1/6: 1 = 1/2 + 1/3 + 1/6.

6² ends in 6. The other digit with this property is 5.

Every prime number greater than 3 is of the form 6n ± 1.

Any number of the form 6n - 1 has two factors whose sum is divisible by 6.

6 is the 3rd triangular number, and the only triangular number, apart from 1, with less than 660 digits whose square (36) is also triangular.

The following property is due to Iamblichus. Take any 3 consecutive numbers, the largest divisible by 3. Add them, and add the digits of the result, repeating until a single number is reached. That number will be 6.

The second and third Platonic solids, which are duals of each other, the cube and the octahedron, have 6 faces and 6 vertices respectively.

The first, the tetrahedron, has 6 edges.

Regular polytopes

There are 6 regular polytopes. They are the analogues in 4 dimensions of the regular polyhedra in 3 dimensions and the regular polygons in 2 dimensions.

Each polytope has vertices, edges, faces and also cells. Two of them are self-dual, the others form two dual pairs.

6 equal circles can touch another circle in the plane.

One of the 3 regular tesselations of the plane is composed of regular hexagons.

Pappus discussed the practical intelligence of bees in constructing hexagonal cells. He supposed that the cells must be contiguous, to allow no foreign matter to enter, must be regular, and therefore either tri- / Page 69 / angular, square, or hexagonal, and concluded that bees knew that a hexagon, using the same material, would hold more than the other shapes.

Pappus, claiming that man has a greater share of wisdom than the bees, then went on to show that of all regular figures with equal perimeter, the one with the larger number of sides has the larger area, the circle being the limiting maximum.

Kepler discussed the 6-fold symmetry of snowflakes, and attempted to explain it by considering the close packing of spheres in a hexagonal array.

Pascal discovered in 1640 at the age of 16 his theorem of the Mystic Hexagram. If any six points are chosen on a conic section, labelled 1,2, 3,4, 5,6, then the intersections of the lines 12 and 45, 34 and 61, 56 and 23, will lie on a straight line.

Brianchon enunciated the dual theorem, in which the 6 original points are replaced by 6 tangents to the conic.

6.283 185 . . .

2n

The ratio of the circumference to a radius of a circle. The number of radians in a complete circle.

/ Page 70 /

7

7 days in a week, and therefore associated with 14 and with 28 days in a lunar month.

The 4th prime number, and the first of the form 6n + 1.

The start of an arithmetical progression of six primes: 7, 37, 67, 97, 127,157.

7 and 11 are the first pair of consecutive primes different by 4.

The 3rd Mersenne number, 7 = 2³ - 1, and the second Mersenne prime, leading to the second perfect number.

The first number that is not the sum of at most 3 squares. The sequence of such numbers continues, 15 23 28 31 39 47 55 60 ...

7 = 3! + I.n! + 1 is prime for n = 1,2,3,11,27,37,41,73,77,116, 154,320, 340, 399, 427, and no other values below 546.

Brocard's problem. When is n! + 1 a square? The only known solutions are n = 4, 5 and 7: 7! + 1 = 5041 = 71².

The Fermat quotient

is a square only when p is 3, or 7.

Lame proved in 1840 that Fermat's equation, x⁷ + y⁷ = Z⁷ has no solutions in integers.

If a, b are the shorter sides of a Pythagorean triangle, then 7 divides one of a, b, a - b or a + b.

Because 7² falls short of 50 by only 1, 7 was called by the Greeks, the rational diagonal of a square of side 5.

All sufficiently large numbers are the sum of 7 positive cubes.

To test if a number is divisible by 7: multiply the left-hand digit by 3 and add the next digit. Repeat as often as necessary. If the final answer is divisible by 7, so is the original number.

Alternatively, start by multiplying the right-hand digit by 5 and adding the adjacent digit. Repeat as before.

7 numbers are sufficient to colour any map on a torus. Surprisingly, this was known before the 4-colour conjecture was solved for plane maps.

At least 7 rectangles are required if a rectangle is to be divided into smaller rectangles no one of which will fit inside another. The smallest rectangle that can be tiled 'incomparably' is 13 by 22.*

At least 7 rectangles are also required to divide a rectangle into smaller rectangles of different shape but equal area.

*A. C. C. Yao and E. M. Reingold, Journal of Recreational Mathematics, vol. 8.

/ Page 71 / An obtuse-angled triangle can be divided into not less than 7 acute- angled triangles.

There are 7 basically different patterns of symmetry for a frieze design.

The regular 7-gon is the smallest that cannot be constructed by ruler and compass alone.

7 is the smallest prime the period of whose reciprocal in base 10 has maximum length. 1/7 = 0.142857142857. . . (See 142.857.)

The problem of St lves

This Mother Goose rhyme is well known:

'As I was going to St Ives, I met a man with seven wives. Every wife had seven sacks, and every sack had seven cats, every cat had seven kittens. Kittens, cats, sacks and wives, how many were going to St Ives?'

Problem 79 of the Rhind papyrus, written by the scribe Ahmes, which dates from about 1650 BC, concerns:

The resemblance is remarkable. Moreover, there is a connecting link, of sorts. Leonardo of Pisa, called Fibonacci, in his Liber Abaci (1202 and 1228) also includes the same problem. Pierce comments that it seems to be of the same origin as the House that Jack built, and that Leonardo uses the same numbers as Ahmes and makes his calculations in the same way.

It is tempting to suppose that this problem is indeed more than 3500 years old, and has survived essentially unchanged throughout that time.

8

The second cube: 8 = 2³ . The only cube that is one less than a square 8 = 3² - 1 and the only power that differs by 1 from another prime power.

/ Page 72 / The sixth Fibonacci number, and the only Fibonacci number that is a cube, apart from 1.

The number of parts into which 3 dimensional space is divided by 3 general planes.

There are 8 notes in an octave,

The first number in English alphabetic sequence.

It is possible to place the maximum 8 queens on a chessboard, so that no queen attacks any other, in 12 essentially different ways.

8 times any triangular number is 1 less than a square.

A number is divisible by 8 if the number formed by its last three digits is divisible by 8.

Magic cubes

Perfect magic cubes, in which all the rows, columns and diagonals of every layer, plus the space diagonals through the centre, sum to the same total.are impossible for orders 3 (3 x 3 x 3) and 4 (4 x 4 x 4). It is not known if such cubes can exist for orders 5 and 6.

Magic cubes do exist for order 8. The first was privately published in 1905, a method of construction was again discovered in the late 1930s, and in 1976 Martin Gardner published an example constructed by Richard Myers.

Myers discovered how to construct vast numbers of them by super- imposing three Latin cubes and using octal notation when he was a 16- year-old schoolboy.

Soon after Gardner reported on Myers's discovery, Richard Schroeppel and Ernst Straus independently found order- 7 magic cubes.

Martin Gardner, Scientific American, January 1976.

The octal system

8 is the base of the octonary, octenary, or octal system.

Emmanuel Swedenborg, the Danish philosopher, wrote a book advocating base 8. It has much of the simplicity of the binary system.

All its factors are powers of 2, yet numbers of a reasonable size do not take an absurdly large number of digits to express. 100 in base 10 is 144 in base 8 and 1100100 in binary. The binary is much harder to remember (always a great disadvantage for practical purposes) and longer, though it can be obtained instantly from the octal 144 by replacing the digits by their binary expression. 1-4-4 becomes 1-100-100 or 1100100.

Arguments for changing to base 8 completely are weaker than for changing to duodecimal. But because of connection with binary, it has /Page 73 / been used extensively in computers, though since the IBM 360 series was introduced in the early 1960s, using base 16 (hexadecimal), it has fallen out of favour.

A deltahedron is a polyhedron all of whose faces are triangular.

There are an unlimited number of them, since any deltahedra has exposed faces to which another triangular pyramid can be attached. However, only 8 of them are convex. 3 of these are the regular tetrahedron, octahedron and icosahedron. 2 more are a pair of tetrahedra glued face to face, and a pair of pentagonal pyramids glued face to face,

The octahedron has 8 triangular faces, and 6 vertices and 12 edges, making it the dual of the cube, which has 8 vertices, 6 faces and 12 edges.

Thus, if the 6 mid-points of the faces of a cube are joined together, they form an octahedron. Conversely, the 8 mid-points of the faces of an octahedron join to form a cube.

9

The third square, and therefore the sum of two consecutive triangular numbers: 9 = 3 + 6,

Written as '100' in base 3.

The first odd prime power, and with 8 the only powers known to differ by 1.

The only square that is the sum of two consecutive cubes: 9 = 1³ + 2³. The 4th Lucky number, and the first square Lucky number apart from 1.

9 = I! + 2! + 3!

The smallest Kaprekar number apart from 1: 9² = 81 and 8 + 1 = 9.

9 is subfactorial 4.

There are 9 regular polyhedra, the 5 Platonic solids and the 4 Kepler- Poinsot stellated polyhedra.

9 is the smallest number of distinct integral squares into which a rectangle may be divided. The smallest solution is 32 by 33 and the squares have sides 1,4,7,8,9,10,14,15, and 18.

The Feuerbach, or nine-point circle

In 1820 Brianchon and Poilcelet proved that the feet of the altitudes, the mid-points of the sides and the mid-points of the segments of the altitudes from the vertices to their point of intersection, all lie on a circle.

/ Page 74 /

Feuerbach proved two years later that this circle also touches the inscribed and three escribed circles of the triangle, and in consequence it is often known as the Feuerbach circle.

Because 9 is one less than the base of our usual counting system, there is a simple test for divisibility by 9. 9 divides a number if and only if it divides the sum of the number's digits.

Arithmetical sums may be checked by the process called 'casting out nines'. This came to Europe from the Arabs, but was probably an Indian invention. Leonardo of Pis a described it in his Liber Abaci. Each number in a sum is replaced by the sum of its digits. (Originally it was replaced by the remainder on dividing by 9, which is a long way round of coming to the same result.)

If the original sum is correct, so will the same sum be when performed with the sums-of-digits only.

Which fits better, a round peg in a square hole or a square peg in a round hole? This can be interpreted as, which is larger, the ratio of the area of a circle to its circumscribed square, or the area of a square to its circumscribed circle?

/ Page 80 /

The Pythagoreans

Pythagoras and his disciples taught that everything is Number. Numbers to them meant strictly whole numbers, integers. Fractions were considered only as ratios between integers.

The Greeks distinguished between logistike (whence our term logistics), which meant numeration and computation, and arithmetike, which was the theory of numbers themselves.

It was arithmetike that Plato, a convinced Pythagorean, insisted should be learned by every citizen of his ideal Republic, as a form of moral instruction. It was a profound shock to their philosophy when (square root) /2 was discovered to be not the ratio of two integers, although it was undoubtedly a length and therefore, to the Greeks who thought of numbers geometrically, a number or ratio of numbers.

Pythagoras himself or his disciples discovered that harmony in music corresponded to simple ratios in numbers. Indeed, it was this discovery that provided the earliest support for their doctrine. Aristotle records that, 'They supposed the elements of number to be the elements of all things, and the whole heaven to be a musical scale and a number.'

The octave corresponds to the ratio 2: 1 because if the length of a musical string is halved, it sounds one octave higher. The ratio 3: 2 corresponds to the fifth and 4:3 to the fourth.

Somewhat less harmonious intervals were represented by rather larger numbers. A single tone was the difference between a fifth and a fourth, and was therefore 9: 8, which is 3: 2 divided by 4: 3.

(The problem of constructing a complete scale is very complex, and / Page 81 / has engaged the efforts of musicians to the present day. All solutions involve approximation. It is not possible for example for a fixed scale, such as a piano possesses, to include all the perfect fifths and fourths that the performer would like. The violinist has an advantage here over the pianist. The solution that divides the octave into 12 equal tones gets none of them perfectly correct.)

The basic ratios could be represented in the sequence 12: 9: 8: 6 and the sum of these numbers, 35, was called harmony.

More commonly, the Pythagoreans thought of these ratios as involving only 1, 2, 3 and 4, whose sum is 10, which is the base of our counting system. How elegantly everything fits together! No wonder they felt confirmed in their diagnosis of the vital significance of Number.

The number 10 can also be represented as a triangle, which they called the tetraktys. To the Pythagoreans it was holy, so holy that they even swore oaths by it.

Later Pythagoreans described many other tetraktys. Magnitude, for example, comprised point, line, surface and solid.

The primitive aspects of Pythagorean belief died out very slowly. Their musical discoveries did not die out at all. They were true science, two thousand years before modern science displayed the whole numbers in the chemist's Periodic Table or the physicist's model of the atom.

Precisely because music was for so long a unique example of genuine numbers-in-science, it had an overwhelming effect. Leibniz wrote, 'Music is a secret arithmetical exercise and the person who indulges in it does not realize that he is manipulating numbers.'

That is not quite correct. Early classical composers, before the advent / Page 82 / of Romanticism, were often quite deliberate in their use of mathematical patterns to structure their music.

Unlike the Greeks, we are not limited to the whole numbers, and today science often seems to be soaked in rational approximations, rational results from experimental observation.

Yet underneath the complexity of modem science, the integers may still occupy a central role. Daniel Shanks gives many examples of their role in modem science. To relate just one of his examples, why is the force of gravity at double the distance reduced by a factor of 4? Why is the factor apparently 4 exactly, rather than 4 approximately? Probably because we live in a space of exactly 3 dimensions.

The Pythagoreans' faith .in the whole numbers may be vindicated yet.

Daniel Shanks, Solved and Unsolved Problems in Number Theory, vol. I, Spartan Books, 1962.

11

The 5th prime.

The smallest repunit, a number whose digits are all units.

11, like all repunits, is divisible by the product of its digits.

Because 11 = 10 + 1, there is a simple test for divisibility by 11. Add and subtract the digits alternately, from one end. (Either end may be chosen as the starting point.)

If the answer is divisible by 11, so is the number.

This is equivalent to adding the digits in the odd positions, and in the even positions, and subtracting one answer from the other.

11 appears as a factor, and a multiple, though not by itself, in the imperial system of measuring length. 5 yards was one rod, pole or perch; 22 yards is a chain; 220 yards a furlong; and 1760 = 11 x 160 yards makes 1 mile.

11 is the only palindromic prime with an even number of digits. Given any 4 consecutive integers greater than 11, there is at least one of them that is divisible by a prime greater than 11.

The world we live in is apparently 3-dimensional, or 4-dimensional when time is counted as an extra dimension.

According to the latest physical theory of supersymmetry, space is most easily described as 11-dimensional.

Seven of the dimensions are 'curled up on themselves'. Their physical effects would be directly observable only on a still inaccessible scale billions of times smaller even than that of subatomic particles.

Another bizarre but spectacular idea related to supersymmetry is that / Page 83 / the basic units of both matter and force are phenomena called strings, and that the various fundamental particles correspond to the different ways these strings vibrate, like the harmonics of a violin.

Bryan Silcock, 'The Cosmic Gut', The Sunday Times, 24 March 1985.

/ Page 96 /

17

The 3rd Fermat prime: 17 = 2^{2 2} + 1.

Gauss proved at the age of 18 that a regular polygon can be constructed with the use only of a straight edge and compasses only if the number of sides is the product of distinct Fermat primes, of the form 2²? + 1.

It is possible therefore to construct a regular 17 -gon with ruler and compasses only.

The period of 1/17 is of maximal length, 16:

1/17 = 0.0588235294117647.

17 is the first sum of two distinct 4th powers: 17 = 1⁴ + 2⁴.

17 is equal to the sum of the digits of its cube, 4913. The only other such numbers are 1,8, 18, 26 and 27, of which three are themselves cubes.

Choose numbers a, b, c . . . in the interval (0, 1) so that a and b are in different halves of the interval, a, b and c are in different thirds, a, b, c and d are in different quarters and so on.

Not more than 17 such numbers can be chosen.

There are 17 essentially different symmetry patterns for a wallpaper design.

17 is the highest number whose square root was proved irrational by Theodorus.

According to Plutarch, 'The Pythagoreans also have a horror of the number 17. For 17 lies halfway between 16... and 18... these two being the only two numbers representing areas for which the perimeter (of the rectangle) equals the area.'.*

n² + n + 17 is one of the best known polynomial expressions for primes. Its values for n = 0 to 15 are all prime, starting with 17 and ending with 257.

The only known prime values for which p^q - 1 and q^P - 1 have a common factor less than 400000 are 17 and 3313. The common factor is 112643.**

* Van de Waerden. Science Awakening, Oxford University Press. New York, 1971.

** N. M. Stephens, 'On the Feit-Thompson Conjecture., Mathematics of Computation, vol. 25.

/ Page 97 /

18

18 = 9 + 9 and its reversal, 81 = 9 x 9.

This pattern works in any base. For example, in base 8: 7 + 7 = 16 and 7 x 7 = 61.*

The cube and 4th powers of 18 use all the digits 0 to 9 once each:

18³ = 5832 and 18⁴ = 104976.

18 is equal to the sum of the digits of its cube: 18³ = 5832.

19

The 3rd number whose decimal reciprocal is of maximum length, in this case 18: 1/19 = 0.052631 578947 368421.

There is a simple test for divisibility by 19. 100a + b is divisible by 19 if and only if a + 4b is.

19 is the 3rd centred hexagonal number: 19 = 1 + 6 + 12.

There is only one way in which consecutive integers can be fitted into a magical hexagonal array, that is, so that their sums in all three directions are all equal. The numbers 1 to 19 can be so arranged, a fact first discovered by T. Vickers.

19! - 18! + 17! - 16! + . . . + 1 is prime. The only other numbers with this property are 3, 4, 5, 6, 7, 8, 10 and 15. [Guy]

All integers are the sum of at most 19 4th powers.

20

The sum of the first 4 triangular numbers, and therefore the 4th tetra- hedral number: 20 = 1 + 3 + 6 + 10.

An icosahedron has 20 faces and its dual, the dodecahedron, has 20 vertices.

20 is the second semi-perfect, or pseudonymously pseudoperfect number, because it is the sum of some of its own factors:

20 = 10 + 5 + 4 + 1.

The smallest semi-perfect is 12, which is also the first abundant. number. The next are 20, 24 and 30.

The vigesimal system

20 has a special significance in many systems of counting and of weights and measures.

Base 20, called vigesimal, was used by the Mayan astronomers and calendar makers whose culture flourished from the 4th century AD. Their system was positional and included a zero, centuries before the appearance of Indian numerals in Europe.

*D. Y. Hsu, Journal of Recreational Mathematics, vol. 10.

/ Page 98 /

20 occurs in the old English coinage in '20 shillings in the pound' and in the imperial system of weights and measures.

20 is a score, and ages in biblical language are often expressed in scores: 'The days of our years are threescore and ten; and if by reason of strength they be fourscore years, yet is their strength labour and sorrow.'

'A score' or 'scores' survives as an expression for a largish number.

21

The 6th triangular number, and therefore the total number of pips on a normal dice.

If a square ends in the pattern xyxyxyxyxy, then xy is either 21,61 or 84.

The smallest example is: 508853989² = 258932382121212121.

21 is the smallest number of distinct squares into which a square can be dissected.

The side of the dissected square is 112.

/ Page 118 /

In Pascal's triangle, each number is the sum of either of the diagonals starting immediately above it, and taking the long way to the edge: for example, 35 = 15 + 10 + 6 + 3 + 1. (So the sum of the first 5 triangular numbers is 35.)

The first few rows of Pascal's triangle may give the impression that almost all its entries are different, apart from the left-right symmetry and the edge units. This is not so, as the appearance of three 6s and four 10s might suggest. (See 3003.)

36

The 8th triangular number, thought of by the Greeks as also the sum of the first 4 even numbers and first 4 odd numbers.

It is also square, and the first number after 1 to be both square and triangular.

The numbers that are both square and triangular are beautifully related to the best approximations to (square root) / 2:

and so on.

In each case the factors of the root are the numerator and denominator of the next approximation to (square root) / 2.

Because its square root is the 3rd triangular number, it is also the sum of the first 3 cubes: 36 = 1³ + 2³ + 3³.

1-6-36 is the first set of triangular numbers in geometrical progression.

36 is the largest 2-digit number divisible by the product of its digits. Every sequence of 7 consecutive numbers greater than 36 includes a multiple of a prime greater than 41.

H. Gupta, Selected Topics in Number Theory, Abacus Press, Tunbridge Wells, 1980.

37

Any 3-digit multiple of 37 remains a multiple when its digits are cyclically permuted.

Every number is the sum of at most 37 5th powers.

The 4th centred hexagonal number, obtained by arranging hexagonal layers of points around a central point.

/ Page 127 /

60

The base of a sexadecimal system of counting.

The Sumerians as early as 3500 BC had a decimal system for business purposes and a sexadecimal system used by a small number of experts, based on 10s and 6s: 1, 10, 60, 600, 3600, 36000. . .

The Babylonians used this sexadecimal system for mathematical and astronomical work.

Systems based on 60 benefit from the many factors of 60. They have the advantages of a duodecimal system, and more.

In astronomy, the very ancient division of the Zodiac into 12 parts fits a sexadecimal system very well, and does not fit a decimal system at all.

The division of the circle into 360 degrees, and the division of degrees into 60 and 3600 parts originated among Babylonian astronomers a few centuries B C.

We still divide an hour of time or an angle of one degree into 60 minutes and each minute into 60 seconds. These are the only common measurements that have not been metricated.

60 degrees is the interior angle of an equilateral triangle.

Highly composite numbers

The 8th 'highly composite' number, defined by Ramanujan as a number that, counting from 1, sets a record for the number of its divisors.

60 = 2² x 3 x 5 is the first number with 12 divisors.

/ Page 128 /

The sequence of 'highly composite' numbers starts: 2 4 6 12 24 36 48 60 120 180 240 360 720 840 1260 1680 2520 5040 ...

G. H. Hardy, Collected Papers of S. Ramanujan, Cambridge University Press, Cambridge, 1927.

61

1318820881² = 1739288516161616161 (See also 21.)

63

Kaprekar's process for 2-digit numbers leads to the cycle: 63-27-45-9- 81-63 . . . In this cycle, 9 must be read as the 2-digit number 09.

For example, starting with 5 and 3: 53 - 35 = 18; 81 - 18 = 63, entering the cycle. Or, starting with 9 and 3: 93 - 39 = 54; 54 - 45 = 9, entering the cycle at a different point.

64

The second 6th power, after 1, and also a square and a cube:

64 = 4³ = 8² = 2⁶.

It is therefore represented by 100 in octal and by 1,000,000 in binary.

The smallest number with 6 prime factors. The next smallest are 96, 128 (which has 7) and 144.

Being a cube, it is the sum of consecutive centred hexagonal numbers: 1+ 7 + 19 + 37 = 64.

Fermat's Little Theorem says that if p is prime then a^{(P - 1)} - 1. is divisible by p, provided a is not divisible by p.

For every prime p, there are values of a such that a^{(p - 1)} - 1 is actually divisible by p².

The smallest such value for p = 3 is 8² = 64: 64 - 1 is divisible by 3² = 9.

For p = 5, the next prime, the smallest solution is 7⁴ - 1, which is divisible by 25.

65

The second number to be the sum of two squares in two ways:

65 = 8² + 1² = 7² + 4²,

65 is the magic constant in a 5 by 5 magic square.

66

The sum of the divisors of 66, including 66 itself, is a square: 1 + 2 + 3 + 6 + 11 + 22 + 33 + 66 = 144 = 12².

/ Page 129 /

The sequence of numbers with this property starts: 3 22 66 70 81 ...

69

The only number whose square and cube between them use all the digits 0 to 9 once each: 69² = 4761 and 69³ = 328509.

70

The sum of its divisors, including 70 itself, is a square, 144.

Weird numbers

The smallest weird number. A number is called weird if it is abundant without being the sum of any set of its own divisors. The factors of 70 are 1, 2, 5, 7, 10, 14 and 35, which sum to 74, so it is abundant, but no set of them sum to 70.

Weird numbers are rare. The only ones below 10,000 are 70, 836, 4030, 5830, 7192, 7912 and 9272.

Note that they are all even. It is not known whether an odd weird number exists. Professor Pal Erdos, who has the charming habit of offering money for the solutions to mathematical challenges, was offering, in 1971, $10 for the first example of an odd weird number, or $25 for a proof that none exist. This shows a nice judgement of the relative value of a counter example and a proof!

71

71² = 7! + 1. This is the largest known solution to Brocard's problem.

71³ = 357911. The digits are the odd numbers 3 to 11 in sequence.*

The numbers 5, 71 and 369119 are the only numbers less than 2,000,000 that divide the sum of the primes less than them. **

72

(calculation ommitted)

This is the smallest set of four numbers in arithmetical progression whose (symbol ommitted) values are equal.

The next two 4-term arithmetical progressions with equal (symbol ommitted) values start at 216 and 76236 and each also has common difference, 6.***

72⁵ = 19⁵ + 43⁵ + 46⁵ + 47⁵ + 67⁵ is the smallest 5th power equal to the sum of 5 other 5th powers.

* James Davies, Journal of Recreational Mathematics, vol. 13.

** Ibid., vol. 14.

*** M. Lal and P. Gillard, Mathematics of Computation, vol. 26.

/ Page 130 /

73

All integers can be represented as the sum of at most 73 6th powers.

76

76² = 5776, which ends in the digits 76, which is therefore called automorphic.

The only other 2-digit automorphic number below 100 is 25.

Automorphic numbers are related to multiples of powers of 10. For example, 76 x 75 = 57 x 10².

77

Every number greater than 77 is the sum of integers, the sum of whose reciprocals is 1.

For example, 78 = 2 + 6 + 8 + 10 + 12 + 40 and 1/2 + 1/6 + 1/8 + 1/10 + 1/12 + 1/40 = 1.

R. L. Graham, 'A Theorem on Partitions', Journal of the Australian Mathematical Society, 1963; quoted in Le Lionnais, 1983.

79

The smallest number that cannot be represented by less than 19 4th powers: 79 = 15 x 1⁴ + 4 x 2⁴.

81

81 = 3⁴

The sum of the divisors of 81 is 121, a square.

The fraction 1/81 = 0.012345679 012345679 012 . . .

This pattern occurs because 81 = 9² and 9 is 1 less than 10, the base of the decimal system.

In another base, 6 for example, the reciprocal of (6 - 1)² is 1/41 = 0.012350123501235...

81 is the only number whose square root is equal to the sum of its digits, apart from the trivial 0 and 1.

81 is both square and heptagonal.

Write the natural numbers in groups, like this:

1

2,3

4,5,6

7,8,9,10

11,12,13,14,15 ...

Delete every second group, The sum of the first remaining n groups is then n⁴.* For example,

1 + 4 + 5 + 6 + 11 + 12 + 13 + 14 + 15 = 81 = 3⁴

More straightforward is this pattern:

* Dov Juzuk, Scripta Mathematica, 1939.

/ Page 131 /

and so on.* The number of terms in each sequence is 1, 3, 9, 27. . .

84

Little is known of the life of Diophantus. This verse from The Greek Anthology purports to give his age, which turns out to be 84.

This tomb holds Diophantus. Ah, how great a marvel! The tomb tells scientifically the measure of his life. God granted him to be a boy for one-sixth of his life, and adding a twelfth part to this, he clothed his cheeks with down. He lit him the light of wedlock after a seventh part, and five years after his marriage he gave him a son. Alas, late-born wretched child! After attaining the measure of half his father's life, chill Fate took him. After consoling his grief by the study of numbers for four years, Diophantus ended his life.

85

The sum of two squares in two ways: 85 = 9² + 2² = 7² + 6².

88

88 is itself a repeated digit, and its square ends in a repeated digit: 88² = 7744.

89

89 and 97 are the first pair of consecutive primes differing by 8.

Double 89 and add 1: repeat, to get a sequence of 6 primes,

89 179 359 719 1439 2879.

This is the smallest such 6-prime sequence. **

Add the squares of the digits of any number: repeat this process, and eventually the number either sticks at 1, or goes round this cycle: 89-145-42-20-4-16-37-58-89.. .

89 and 98 are the 2-digit numbers that require most reversals-and- adding to become palindromes. They each require 24 steps.

89 is the 11th Fibonacci number, and the period of its reciprocal is generated by the Fibonacci sequence: 1/89 = 0.11235 . . .

90

The number of degrees in a right angle.

* M. N. Khatri, Scripta Mathematica, vol. 20.

** Journal of Recreational Mathematics, vol. 13.

/ Page 140 /

153

153 = 1! + 2! + 3! + 4! + 5!

When the cubes of the digits of any 3-digit number that is a multiple of 3 are added, and then this process is repeated, the final result is 153, where the process ends, because 153 = 1³ + 5³ + 3³.

The other 3-digit numbers that equal the sum of the cubes of their own digits are 370, 371 and 407.

These pairs switch from one to the other in a 2-cycle: 136 and 244; 919 and 1459.

There are two cycles of length 3: 55-250-133 and I 60-217-352.

When G. H. Hardy wished, in his book A Mathematician's Apology, to give examples of mathematical theorems that were not 'serious', he chose two examples, 'almost at random, from Rouse Ball's Mathematical Recreations'.

The first was the fact that 8712 and 9801 are the only 4-digit numbers that are multiples of their reversals.

The second was the fact that, apart from 1, there are just 4 numbers that are the sums of the cubes of their digits, those mentioned above.

Hardy commented,

These are odd facts, very suitable for puzzle columns and likely to amuse amateurs, but there is nothing in them that appeals to the mathematician. The proofs are neither difficult nor interesting - merely a little tiresome. The theorems are not serious; and it is plain that one reason. . . is the extreme speciality of both the enunciations and the proofs, which are not capable of any significant generalization.

As any critic might have remarked of Euler's solution of the Bridges / Page 141 / of Konigsberg problem, or of Euler's dabbling in Magic Squares. The existence or non-existence of significant generalizations would appear to be a contingent fact, not susceptible to proof by G. H. Hardy.

As an almost certainly less interesting fact I would suggest, 'The 10,000,000 digit of (symbol ommitted) is a 7,' mentioned by Keith Devlin, though the supposed lack of interest in this fact is still a matter of contingent fact, and no more.

In the New Testament the net that Simon Peter drew from the sea of Tiberias held 153 fishes. This was inevitably interpreted numerologically by the early Church fathers, especially St Augustine.

153 is the 17th triangular number and therefore already significant. But what is special about 17 itself? It is the sum of 10 for the Ten Commandments of the Old Testament to 7, for the Gifts of the Spirit in the New Testament.

This was a common means of combining two influences, just as the Pythagoreans associated 5 with marriage because 5 = 2 + 3 and those numbers are female and male respectively.

W. E. Bowman, a modern writer with more humour and less reverence, introduces the number 153 on numerous occasions into his novel The Ascent of Rum Doodle. It appears as the height of the ship above sea level, the speed of a train chugging through the foothills of the Himalayas, the number of porters to be hired for the ascent, and the depth of a crevasse, among other things.

154

154! + 1 is prime.

/ Page 142 /

180

The number of degrees in a half-circle, and the number of degrees Fahrenheit between the freezing point of water, 32, and its boiling point, 212.

The sum of the angles of a triangle.

180³ is the sum of consecutive cubes: 180³ = 6³ + 7³ + 8³ + ... + 68³ + 69³. [Beiler]

/ Page 144 /

216

216 = 6³ is the smallest cube that is also the sum of 3 cubes:

216 = 3³ + 4³ + 5³. The next smallest is 9³ = 1³ + 6³ + 8³.

This dissection can be demonstrated physically by dissecting a cube, using only 8 pieces.

216 is the magic constant in the smallest possible multiplicative magic square, discovered by Dudeney.

Plato's number

The famous and notorious number of Plato occurs in an obscure passage in The Republic, viii, 546 B-D, which starts,

But the number of a human creature is the first number in which root and square increases, having received three distances and four limits, of elements that make both like and unlike and wax and wane, render all things conversable and rational with one another.

This is merely the beginning of the passage. It illustrates perfectly both the intimate relationship that Plato, as a Pythagorean, perceived between numbers and the real world, and the difficulty that he had in using the then available language to express himself. Mathematical language was not well developed in Plato's time, and so he often apparently called upon the resources of everyday language. I say 'apparently' because some words in the passage are hardly known in other preserved writings and therefore their meaning is especially difficult to interpret. (The obscurity is not entirely due to our distance from Plato in time. Early Greek commentators also found the passage difficult.)

The whole passage has been analysed in the minutest detail by in- numerable commentators. Two numbers are actually involved and the smaller it is agreed is 216, though this is variously derived. (The larger is 12,960,000.)

The well-known 3-4-5 Pythagorean triangle has area 6. The expression 'three distances and four limits' is supposed to refer to cubing. Adams eventually reaches the conclusion that the number intended in the quoted passage is 216 as the sum of the cubes of the sides of the triangle. However, it has also been deduced as the cube of 2 x 3.

2 and 3 were associated with female and male respectively, and 5 with marriage. 6 also was associated with marriage, being 2 x 3 rather than 2 + 3. Given the Pythagoreans' basic belief in the efficacy of numbers in interpreting the world, it can hardly be denied that such number-theoretic relationships as this support their approach.

J. Adams, The Republic of Plato, Cambridge University Press, Cambridge, 1929. 144

/ Page 145 /

217

The second smallest pseudoprime to base 4 (15 is the smallest).

4²¹⁶ - 1 is divisible by 217 although 217 is not prime but 7 x 31.

219

There are 219 space groups in 3 dimensions. They are the analogues of the 17 basic wallpaper patterns in 2 dimensions, and determine the possible shapes of mineral crystals.

11 of them however come in 2 forms, with a left-hand screw or a right- hand screw. This difference is important in the structure and optical properties of crystals, so from this point of view there are 230 space groups.

220

Amicable numbers

220 and 284 form the first and smallest amicable pair. Each is the sum of the proper divisors of the other: 220 = 2² x 5 x 11 and its proper divisors are 1,2,4,5,10, 11,20,22,44,55 and 110: total 284.

284 = 2² x 71 and its proper divisors are 1, 2,4,71 and 142, totalling 220.

According to Iamblichus, Pythagoras knew of this pair. However, Pythagoras may possibly not be the only ancient wise man to know of amicable numbers. Bible commentators point to Jacob's gift of 220 goats to Esau on their reunion - a friendly gift?

The brilliant Muslim mathematician, astronomer and physician Thabit ibn Qurra described in his Book on the Determination of Amicable Numbers Euclid's rule for perfect numbers, means of constructing abundant and deficient numbers, and the first rule for constructing amicable numbers, from which he deduced Pythagoras' pair, or perhaps more probably, the factors of 220 and 284 suggested the form of his rule:

Find a number, n, greater than 1, that makes these three expressions all prime:

a = 3 x 2ⁿ - 1 b = 3 x 2^{n -1} - 1 c = 9 x 2^{2n -1} - 1

Then the pair 2ⁿ x a x b and 2ⁿ x c will be amicable.

The smaller of any Thabit pair is a tetrahedral number. 220 is the 10th tetrahedral. Lee and Madachy suggest that it may be significant that the first perfect number, 6, equals 1 x 2 x 3; the smallest multiply perfect, 120, is 4 x 5 x 6 and the sum of 220 and 284 is 504 = 7 x 8 x 9. They comment that the Babylonians are known to have constructed tables / Page 146 / of the products of 3 consecutive numbers, which are just 6 times the tetrahedral numbers.

There is an obvious similarity to Euclid's rule for even perfect numbers. However, Thabit's rule does not give all amicable pairs. Indeed, it is one of a number of similar patterns that generate amicable pairs. It is also very difficult to use, because it involves making 3 expressions prime simultaneously. Thabit ibn Qurra himself found no new pair. In fact his rule works for n = 2, 4 and 7, but for no other values below 20,000.

The second pair, 17,296 and 18,416, was discovered by another Arab, Ibn al-Banna. It is Thabit's rule for n = 4. This pair was then re- discovered in 1636 by Fermat who also rediscovered Thabit's rule, as did Descartes who produced a third pair, 9,363,584 and 9,437,056, two years later. This is the Thabit formula for n = 7.

Euler was the first mathematician successfully to explore amicable numbers and find many examples, more than 60. His methods are still the basis for present-day exploration.

Well over a thousand pairs of amicable numbers are now known, including all possible pairs in which the smaller number is less than a million.

The largest, discovered by te Riele, is the pair: 3⁴ x 5 x 11 x 5281¹⁹ x 29 x 89(2 x 1291 x 5281¹⁹ - 1) and 3⁴ x 5 x 11 x 5281¹⁹(2³ x 3³ x 5² x 1291 x 5281¹⁹ - 1), each of 152 digits.

The methods of te Riele also allow him to generate new amicable pairs from old. Applied to a sample of amicable pairs, he obtained more than one 'daughter pair' per 'mother pair', which suggests that perhaps the number of amicable pairs is infinite.

Clearly the greater member of an amicable pair is deficient. Also, neither member of an even-even pair is divisible by 3.

In every case the numbers in a pair are either both even or both odd, though no reason is known why an even-odd pair should not exist.

Every pair also has a common factor. It is not known if a pair of coprime amicable numbers exists. If it does, then even in the most favourable case, in which their product is divisible by 15, that product itself must exceed 10⁶⁷. If they do, they will not of course be constructed on Thabit's pattern, or any similar pattern.

The numbers in every known odd-odd pair are also multiples of 3, so numerous mathematicians have naturally conjectured that this is a general rule.

In 1968 Martin Gardner noticed that the sum of every even pair was divisible by 9 and naturally conjectured that this too was always so. It / Page 147 / isn't, but counter-examples are rather rare; Elvin Lee gave the example 666030256, 696630544, originally discovered by Poulet.

Most amicable numbers have many different factors. Is it possible for a power of a prime, pⁿ to be one of an amicable pair? If it is, then pⁿ is greater than 10¹⁵⁰⁰ and ⁿ is greater than 1400.

A generalization of amicable pairs is amicable triplets, in which the proper divisors of any one number sum to the sum of the other two. Beiler gives this example: 2⁵ x 3 x 13 x 293 x 337; 2⁵ x 3 x 5 x 13 x 16561; 2⁵ x 3 x 13 x 99371.

E. J. Lee and J. Madachy, 'The History and Discovery of Amicable Numbers', parts 1 and 2, Journal of Recreational Mathematics, vol. 5.

/ Page 153 /

365.2422

The calendar

The approximate number of days in a year, equal to 365 days 5 hours 48 minutes and 46.08 seconds.

This is the time taken for the earth to make one revolution of the sun. Every civilization has related it to the period of the moon's phases, for example the time between two new moons, which is approximately 29.530588 days, or 29 days 12 hours 44 minutes and 2.8 seconds.

Unfortunately, the relation cannot be a very simple one. It is coincidental that the length of the year in days is so close to the very round number, 360, which happens to be very close to 12 times the period of the moon.

Such coincidences are helpful, but not enough, and immense ingenuity has been devoted to accounting for the differences.

In the Julian calendar the ordinary years have 365 days but every year whose number is divisible by 4 has an extra day, the 29th February, making a total of 366 days. The average Julian year has therefore 365.25 days and is one day out approximately every 128 years.

The Gregorian calendar, which is used today in most parts of the world, is a small but significant improvement on the Julian. All years divisible by 100 are ordinary years, not leap years, with the exception of years divisible by 400, which remain leap years. The Gregorian calendar contains one too many days every 3333 years, and so will not require adjustment until long after we are all dead.

In the Soviet Union, however, they use an even more accurate calendar, introduced in October 1923. All years are ordinary years except those which when divided by 9 leave either 2 or 6 as remainder. This calendar contains one day too many after 45,000 years.

The Julian and Gregorian calendars are based on the length of the year and therefore on the sun. Given any day of the year, we can tell fairly accurately the position of the sun in the sky, but not the position of the moon.

The Muslim calendar in contrast gives the moon precedence, It has 12 months of alternately 30 and 29 days. In a leap year the last month has an extra day. The ordinary year has only 354 days and a leap year 355 days, so the start of the Muslim year moves steadily through the Gregorian year, and conversely.

The Jewish year is a combination of solar and lunar years. The basic year is a lunar year of 12 months that are alternately of 30 and 29 days, but when the error amounts to a full month, a 13th month is inserted / Page 154 / into that year. This makes it the most complicated by far of all calendars.

The complications that are introduced when the solar year and the lunar month are considered together are well illustrated by the manner in which the date of Easter, which depends on the position of the moon, jumps around in the Christian year. The great Karl Friedrich Gauss demonstrated his insight into numbers by constructing simple formulae for calculating the date of the Christian Easter festival, and also, which is even more difficult, the date of the Jewish festival of the Passover.

W. A. Schocken, The Calculated Confusion of Calendars, Vantage Press, New York, 1976.  

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512-73

Numerology

512-73 is the Dewey Decimal classification, under the general class '510 mathematics' for 'number theory: analytic'.

When Martin Gardner wrote The Numerology of Dr Matrix, the Dewey classification for 'number theory' was, as he pointed out, 512.81, whose two halves are respectively 2⁹ and 9².

No doubt because this trivial piece of numerology has been found out, the authorities have since changed 'number theory' to 512.7 and given this new number, 512.73, to analytic number theory, whose first classification is, significantly, transcendental numbers. I shall now illuminate the profound significance of 512.73 for the benefit of the uninitiated.

First, I subtract it from 666, the Number of the Beast in the Book of Revelation: 666 - 512.73 = 153.27.

Behold! The same digits appear, but rearranged, symbolizing the effect of removing evil from the world. The first number is now 153, the number of fishes hauled from the sea by Peter, which was so eloquently interpreted by St Augustine. The second number is now the sacred number 3, raised to its own power. 153 is also associated with the sacred 3. Not only is its sum of digits equal to 9, which is 3 times itself, but it is the sum of the 3rd power of its own digits. The significance of 3 appears in the Dewey Decimal System. Divide the Number of the Beast by 3, and you obtain 222, the classification of the Old Testament. Add 3, and you obtain 225, the New Testament. Add 3 again, and you obtain 228, which is the Book of Revelation.

And so on, and on, and on, and on, and on. . .

I trust this illustrates how an hour's worth of jiggery-pokery with a selection of numbers (choose the ones you want, ignore the rest) will produce out of the hat any number you desire. . .

527

For any number n, it is possible to choose at most 6 numbers less than n such that the product of their factorials is a square.

527 is the smallest number that actually requires the maximum 6 numbers to be chosen. [Le Lionnais]

559

This is the largest number less than 4100 that requires 19 4th powers for. its representation.

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666

The 36th triangular number (666 = x 36 x 37) and the Number of the Beast in the Book of Revelation: 'Here is wisdom. Let him that hath understanding count the number of the beast; for it is the number of a man, and his number is six hundred, three score and six.'

A number beloved of occultists, who throughout the ages have used gematria to find the Number of the Beast in the names of their enemies, political or theological.

The fact that some ancient authorities give the number as 616 has not deterred them. With a little ingenuity, both numbers can be found instead of just one.

Peter Bungus made Luther equal to 666, by using the old system, which counts A-I as 1-9, K-S as 10-90, and T-Z as 100-500. Bungus / Page 159 / read Luther's name as Martin Luthera, half German and half Latin, a typical bit of skulduggery, but Bungus was an expert. He wrote a dictionary of numerological symbolism.

666 in Roman numerals is DCLXVI, which has led to the suggestion that this is the origin of 666. It could merely be a way of expressing some large, or vague, number.

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729

9³ and the second smallest cube to be the sum of 3 cubes: 9³ = 1³ + 6³ + 8³.

Since 6³ = 3³ + 4³ + 5³, 9³ is also the sum of 5 cubes.

729 = 3⁶ and therefore is 1,000,000 in base 3.

729 is another mysterious number in Plato's Republic:

. . . if one were to express the extent of the interval between the king and the tyrant in respect of true pleasure he will find on completion of the multiplication that he lives 729 times as happily and that the tyrant's life is more painful by the same distance.

729 was of great significance to the Pythagoreans, being 27². Plato combined the two sequences of powers of 2 and 3 as far as the cubes to form the sequence 1 2 3 4 8 9 27. In this series 27 is the sum of all the preceding members.

C. A. Browne interprets the number in terms of a magic square 27 by 27, whose central cell is occupied by 365, the number of days in the year (729 = 364 + 365).*

1/729 has a decimal period of 81 digits, which can be arranged in groups of 9 digits, reading across each row, in this pattern: **

*W. S. Andrews, Magic Squares and Cubes, Dover, New York, 1960.

**V. Thebault, Scripta Mathematica, vol. 19.

/ Page 161 /

780

780 and 990 are the second smallest pair of triangular numbers whose sum and difference (1770 and 210) are also triangular.

818

818 to 831 is the largest gap between two semi-primes less than 1000.

836

Almost all numbers with palindromic squares seem to have an even number of digits. 836 is the first with an odd number: 836² = 698896. It is also the largest number below 1000 whose square is palindromic.

840

840=2³ x 3 x 5 x 7

It is the number below 1000 with the largest number of divisors: 2⁵ = 32.

854

854² = 729316, a sum that uses all the digits 1-9 once each.

873

873 = I! + 2! + 3! + 4! + 5! + 6!

880

There are exactly 880 magic squares of order 4, provided that all rotations and reflections of the same square are counted as one.

945

The first odd abundant number, discovered by Bachet. It is also semi- perfect.

945 = 3³ x 5 x 7 and its divisors sum to 975.

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Odd abundant numbers are quite rare. There are only 23 of them below 10,000.

981

The only known example of 5 triplets of numbers such that the sums of each triplet are equal and their products also are equal, is: 6, 480, 495; 11, 160,810; 12, 144, 825; 20, 81, 880; 33, 48, 900.

The sum of each triplet is 981, and their common product 1425600. [Guy]

999

The minimum sum of pandigital 3-digit primes, 149 + 263 + 587 = 999.

999² = 998001 and 998 + 001 = 999, so 999, like all numbers whose digits are all 9s, is Kaprekar.

In fact, any multiple at all of 999 can be separated into groups of 3 digits from the unit position, which when added will total 999.

The same principle applies to multiples of 9 99 9999 and so on. 999 = 27 x 37 and so 1/27 = 0.037037 . . . and 1/37 = 0.027027. ..

/ Page 169 /

3600

3600 = 60²

The number of seconds in an hour, or seconds in a degree, or minutes in a full circle.

5040

Factorial 7. 5040 = 7! = 1 x 2 x 3 x 4 x 5 x 6 x 7

In bell ringing, a complete sequence of Stedman Triples contains 7! = 5040 changes, and takes three or four hours to ring.

/ Page 170 /

Plato, in the Laws, suggested that a suitable number of men for an ideal city would be that number which contained the most numerous and most consecutive subdivisions. He decides on 5040, indicating that this number has 59 divisors (apart from itself) and can be divided for purposes of war 'and in peace for all purposes connected with contributions and distributions' by any number from 1 to 10.

Moreover, by merely subtracting two hearths from the total, it is then divisible exactly by 11 also.

/ Page 188 /

9,999,999

Napier's original logarithms were not 'natural', to base e, nor were they based explicitly on exponents. Napier assigned the number 10,000,000 the logarithm 0, and 9,999,999 the logarithm 1. By multiplying repeatedly by 9,999,999/10,000,000 he constructed a sequence of numbers with logarithms 2, 3 . . . and so on.

In the appendix to the 1618 English translation of Napier's original work there is a table of natural logarithms, probably due to William Oughtred who invented the straight and the circular slide rules.

John Wallis in 1685 and Johann Bernoulli in 1694 realized that logarithms could be thought of as exponents.

12,345,679

12,345,679 x 1 = 12,345,679 (digit 8 missing)

12,345,679 x 2 = 24,691,358 (digit 7 missing)

12,345,679 x 3 = 37,037,037

12,345,679 x 4 = 49,382,716 (digit 5 missing)

12,345,679 x 5 = 61,728,395 (digit 4 missing)

12,345,679 x 6 = 74,074,074

12,345,679 x 7 = 86,419,753 (digit 2 missing)

12,345,679 x 8 = 98,765,432 (digit 1 missing)

12,345,679 x 9 = 111,111,111

Note that in each product the sequence 1 to 9, with one digit missing, can be read by cycling through the number, with a suitable repeated jump. For example, 61,728,395 can be read as,

and going round again,

12,960,000

This is the second Geometric Number of Plato, associated with 216, according to many commentators. It has been derived in various ways, for example as 60⁴ or as 4800 x 2700.

/ Page 189 / There was a tradition of a Great Year of Plato, though Plato never mentions it, of 36000 years. At 360 days per year, 36000 years occupies 12,960,000 days.

/ Page 201 /

10⁵¹ [52 digits]

The Sandreckoner

Archimedes in his book The Sandreckoner, which he addressed to Gelon, King of Syracuse, describes his own system of counting immense numbers. He starts with the myriad, which was 10,000, and counts up to a myriad myriads describing these as numbers of the first order.

He then takes 1 myriad myriad, or 100,000,000 in our notation, to be the unit of the numbers of the second order. . . and he continues until he reaches the myriad-myriadth order of numbers.

Archimedes is by no means finished! All the numbers constructed so far are only the numbers of the first period! He continues on his gigantic / Page 202 / construction until he reaches 'a myriad-myriad units of the myriad- myriadth order of the myriad-myriadth period'.

The highest number in his notation would now be expressed as

10^{80.000.000.000.000.000}.

He next proposed to count not merely the number of grains of sand on a seashore, or in the whole earth, but the number of grains of sand required to fill the entire universe.

Assuming that one poppy-head would contain not more than 10,000 grains of sand, and that its diameter is not less than 1/40 of a finger's breadth, and assuming that the sphere of the fixed stars, which was to Archimedes the boundary of the universe, was less than 10⁷ times the sphere exactly containing the orbit of the sun as a great circle. . . the number of grains of sand required to fill the universe turns out to be, in our notation, less than 10⁵¹.

By comparison, Edward Kasner and James Newman in discussing a googol, 10¹⁰⁰, estimate the number of grains of sand on Coney Island at 10²⁰.

This extraordinary achievement by Archimedes is unique within Greek mathematics. The Greeks generally had no interest in numbers outside of some geometrical context. However, to the east, Indian Buddhist mathematicians did construct immense 'towers' of numbers, rising in multiples of 10 or 100, in order to count the atoms 'even in the 3 thousand thousand worlds contained in the universe'. Perhaps Archimedes was inspired by these Indian achievements to construct his own system.

See T. L. Heath, Works of Archimedes, Dover, New York, n.d."