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  3 = 2 + 1, 5 = 22 + 1, 17 = 24 + 1, 257 = 28 + 1, 65537 = 216 + 1;

  and if we care to verify the calculation we easily see that the two large numbers displayed above are 232 + 1 and 264 + 1, also numbers of the sequence. We thus have seven numbers belonging to this sequence and the first five of these numbers are primes, but the last two are not primes.

  Observing how the sequence is composed, we note the “exponents” (the upper numbers indicating what powers of 2 are taken), namely 1, 2, 4, 8, 16, 32, 64, and we observe that these are 1 (which can be written 2°, as in algebra, if we like, for uniformity), 21, 22, 23, 24, 26, 26. Namely, our sequence is 22” 4- 1, where n ranges over 0, 1, 2, 3, 4, 5, 6. We need not stop with n = 6; taking n = 7, 8, 9, . . . , we may continue the sequence indefinitely, getting more and more enormous numbers.

  Suppose we wish now to find out if a particular number of this sequence is a prime. Although there are many shortcuts, and whole classes of trial divisors can be rejected by inspection, and although modern arithmetic limits the kinds of trial divisors that need be tested, our problem is of the same order of laboriousness as would be the dividing of the given number in succession by the primes 2, 3, 5, 7, . . . which are less than the square root of the number. If none of these divides the number, the number is prime. Needless to say the labor involved in such a test, even using the known shortcuts, would be prohibitive for even so small a value of n as 100. (The reader may assure himself of this by trying to settle the case n = 8.)

  Fermat asserted that he was convinced that all the numbers of the sequence are primes. The displayed numbers (corresponding to n = 5, 6) contradict him, as we have seen. This is the point of historical interest which we wished to make: Fermat guessed wrong, but he did not claim to have proved his guess. Some years later he did make an obscure statement regarding what he had done, from which some critics infer that he had deceived himself. The importance of this fact will appear as we proceed.

  As a psychological curiosity it may be mentioned that Zerah Colburn, the American lightning-calculating boy, when asked whether this sixth number of Fermat’s (4294967297) was prime or not, replied after a short mental calculation that it was not, as it had the divisor 641. He was unable to explain the process by which he reached his correct conclusion. Colburn will occur again (in connection with Hamilton).

  Before leaving “Fermat’s numbers” 22n” + 1 we shall glance ahead to the last decade of the eighteenth century where these mysterious numbers were partly responsible for one of the two or three most important events in all the long history of mathematics. For some time a young man in his eighteenth year had been hesitating—according to the tradition—whether to devote his superb talents to mathematics or to philology. He was equally gifted in both. What decided him was a beautiful discovery in connection with a simple problem in elementary geometry familiar to every schoolboy.

  A regular polygon of n sides has all its n sides equal and all its n angles equal. The ancient Greeks early found out how to construct regular polygons of 3, 4, 5, 6, 8, 10 and 15 sides by the use of straightedge and compass alone, and it is an easy matter, with the same implements, to construct from a regular polygon having a given number of sides another regular polygon having twice that number of sides. The next step then would be to seek straightedge and compass constructions for regular polygons of 7, 9, 11, 13, . . . sides. Many sought, but failed to find, because such constructions are impossible, only they did not know it. After an interval of over 2200 years the young man hesitating between mathematics and philology took the next step—a long one—forward.

  As has been indicated it is sufficient to consider only polygons having an odd number of sides. The young man proved that a straightedge and compass construction of a regular polygon having an odd number of sides is possible when, and only when, that number is either a prime Fermat number (that is a prime of the form 22n + l), or is made up by multiplying together different Fermat primes. Thus the construction is possible for 3, 5, or 15 sides as the Greeks knew, but not for 7, 9, 11 or 13 sides, and is also possible for 17 or 257 or 65537 or—for what the next prime in the Fermat sequence 3, 5, 17, 257, 65537, . . . may be, if there is one—nobody yet (1936) knows—and the construction is also possible for 3 × 17, or 5 × 257 × 65537 sides, and so on. It was this discovery, announced on June 1, 1796, but made on March 30th, which induced the young man to choose mathematics instead of philology as his life work. His name was Gauss.

  As a discovery of another kind which Fermat made concerning numbers we state what is known as “Fermat’s Theorem” (not his “Last Theorem”). If n is any whole number and p any prime, then np—n is divisible by p. For example, taking p = 3, n = 5, we get 53 – 5, or 125 – 5, which is 120 and is 3 × 40; for n = 2, p = 11, we get 211 – 2, or 2048 – 2, which is 2046 = 11 × 186.

  It is difficult if not impossible to state why some theorems in arithmetic are considered “important” while others, equally difficult to prove, are dubbed trivial. One criterion, although not necessarily conclusive, is that the theorem shall be of use in other fields of mathematics. Another is that it shall suggest researches in arithmetic or in mathematics generally, and a third that it shall be in some respect universal. Fermat’s theorem just stated satisfies all of these somewhat arbitrary demands: it is of indispensable use in many departments of mathematics, including the theory of groups (see Chapter 15), which in turn is at the root of the theory of algebraic equations; it has suggested many investigations, of which the entire subject of primitive roots may be recalled to mathematical readers as an important instance; and finally it is universal in the sense that it states a property of all prime numbers—such general statements are extremely difficult to find and very few are known.

  As usual, Fermat stated his theorem about np—n without proof. The first proof was given by Leibniz in an undated manuscript, but he appears to have known a proof before 1683. The reader may like to test his own powers on trying to devise a proof. All that is necessary are the following facts, which can be proved but may be assumed for the purpose in hand: a given whole number can be built up in one way only—apart from rearrangements of factors—by multiplying together primes; if a prime divides the product (result of multiplying) of two whole numbers, it divides at least one of them. To illustrate: 24 = 2 × 2 × 2 × 3, and 24 cannot be built up by multiplication of primes in any essentially different way—we consider 2 × 2 × 2 × 3, 2 × 2 × 3 × 2, 2 × 3 × 2 × 2 and 3 × 2 × 2 × 2 as the same; 7 divides 42, and 42 = 2 × 21 = 3 × 14 = 6 × 7, in each of which 7 divides at least one of the numbers multiplied together to give 42; again, 98 is divisible by 7, and 98 = 7 × 14, in which case 7 divides both 7 and 14, and hence at least one of them. From these two facts the proof can be given in less than half a page. It is within the understanding of any normal fourteen-year-old, but it is safe to wager that out of a million human beings of normal intelligence of any or all ages, less than ten of those who had had no more mathematics than grammar-grade arithmetic would succeed in finding a proof within a reasonable time—say a year.

  This seems to be an appropriate place to quote some famous remarks of Gauss concerning the favorite field of Fermat’s interests and his own. The translation is that of the Irish arithmetician H. J. S. Smith (1826-1883), from Gauss’ introduction to the collected mathematical papers of Eisenstein published in 1847.

  “The higher arithmetic presents us with an inexhaustible store of interesting truths—of truths too, which are not isolated, but stand in a close internal connection, and between which, as our knowledge increases, we are continually discovering new and sometimes wholly unexpected ties. A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity upon them, are often easily discoverable by induction, and yet are of so profound a character that we cannot find their demonstration till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simpler meth
ods may long remain concealed.”

  One of these interesting truths which Gauss mentions is sometimes considered the most beautiful (but not the most important) thing about numbers that Fermat discovered: every prime number of the form 4n + 1 is a sum of two squares, and is such a sum in only one way. It is easily proved that no number of the form 4n −1 is a sum of two squares. As all primes greater than 2 are readily seen to be of one or other of these forms, there is nothing to add. For an example, 37 when divided by 4 yields the remainder 1, so 37 must be the sum of two squares of whole numbers. By trial (there are better ways) we find indeed that 37 = 1 + 36, = l2 + 62, and that there are no other squares x2 and y2 such that 37 = x2 +jy2. For the prime 101 we have l2 + 102; for 41 we find 42 + 52. On the other hand 19, = 4 × 5 −1, is not a sum of two squares.

  As in nearly all of his arithmetical work, Fermat left no proof of this theorem. It was first proved by the great Euler in 1749 after he had struggled, off and on, for seven years to find a proof. But Fermat does describe the ingenious method, which he invented, whereby he proved this and some others of his wonderful results. This is called “infinite descent,” and is infinitely more difficult to accomplish than Elijah’s ascent to Heaven. His own account is both concise and clear, so we shall give a free translation from his letter of August, 1659, to Carcavi.

  “For a long time I was unable to apply my method to affirmative propositions, because the twist and the trick for getting at them is much more troublesome than that which I use for negative propositions. Thus, when I had to prove that every prime number which exceeds a multiple of 4 by 1 is composed of two squares, I found myself in a fine torment. But at last a meditation many times repeated gave me the light I lacked, and now affirmative propositions submit to my method, with the aid of certain new principles which necessarily must be adjoined to it. The course of my reasoning in affirmative propositions is such: if an arbitrarily chosen prime of the form 4n + 1 is not a sum of two squares, [I prove that] there will be another of the same nature, less than the one chosen, and [therefore] next a third still less, and so on. Making an infinite descent in this way we finally arrive at the number 5, the least of all the numbers of this kind [4n + l]. [By the proof mentioned and the preceding argument from it], it follows that 5 is not a sum of two squares. But it is. Therefore we must infer by a reductio ad absurdum that all numbers of the form 4n + 1 are sums of two squares.”

  All the difficulty in applying descent to a new problem lies in the first step, that of proving that if the assumed or conjectured proposition is true of any number of the kind concerned chosen at random, then it will be true of a smaller number of the same kind. There is no general method, applicable to all problems, for taking this step. Something rarer than grubby patience or the greatly overrated “infinite capacity for taking pains” is needed to find a way through the wilderness. Those who imagine genius is nothing more than the ability to be a good bookkeeper may be recommended to exert their infinite patience on Fermat’s Last Theorem. Before stating the theorem we give one more example of the deceptively simple problems Fermat attacked and solved. This will introduce the topic of Diophantine analysis, in which Fermat excelled.

  Anyone playing with numbers might well pause over the curious fact that 27 = 25 + 2. The point of interest here is that both 27 and 25 are exact powers, namely 27 = 33 and 25 = 52. Thus we observe that yz = x2 + 2 has a solution in whole numbers x, y, the solution is y = 3, x = 5. As a sort of superintelligence test the reader may now prove that y = 3, x = 5 are the only whole numbers which satisfy the equation. It is not easy. In fact it requires more innate intellectual capacity to dispose of this apparently childish thing than it does to grasp the theory of relativity.

  The equation y3 = x2 + 2, with the restriction that the solution y, x is to be in whole numbers, is indeterminate (because there are more unknowns, namely two, x and y, than there are equations, namely one, connecting them) and Diophantine, after the Greek who was one of the first to insist upon whole number solutions of equations or, less stringently, on rational (fractional) solutions. There is no difficulty whatever in describing an infinity of solutions without the restriction to whole numbers: thus we may give x any value we please and then determine y by adding 2 to this x2 and extracting the cube root of the result. But the Diophantine problem of finding all the whole number solutions is quite another matter. The solution y = 3, x = 5 is seen “by inspection”; the difficulty of the problem is to prove that there are no other whole numbers y, x which will satisfy the equation. Fermat proved that there are none but, as usual, suppressed his proof, and it was not until many years after his death that a proof was found.

  This time he was not guessing; the problem is hard; he asserted that he had a proof; a proof was later found. And so for all of his positive assertions with the one exception of the seemingly simple one which he made in his Last Theorem and which mathematicians, struggling for nearly 300 years, have been unable to prove: whenever Fermat asserted that he had proved anything, the statement, with the one exception noted, has subsequently been proved. Both his scrupulously honest character and his unrivalled penetration as an arithmetician substantiate the claim made for him by some, but not by all, that he knew what he was talking about when he asserted that he possessed a proof of his theorem.

  It was Fermat’s custom in reading Bachet’s Diophantus to record the results of his meditations in brief marginal notes in his copy. The margin was not suited for the writing out of proofs. Thus, in commenting on the eighth problem of the Second Book of Diophantus’ Arithmetic, which asks for the solution in rational numbers (fractions or whole numbers) of the equation x2 + y2 = a2, Fermat comments as follows:

  “On the contrary, it is impossible to separate a cube into two cubes, a fourth power into two fourth powers, or, generally, any power above the second into two powers of the same degree: I have discovered a truly marvellous demonstration [of this general theorem] which this margin is too narrow to contain” (Fermat, Oeuvres, III, p. 241). This is his famous Last Theorem, which he discovered about the year 1637.

  To restate this in modern language: Diophantus’ problem is to find whole numbers or fractions x, y, a such that x2 + y2 = a2; Fermat asserts that no whole numbers or fractions exist such that x3 + y3 = a3, or x4 + y4 = a4, or, generally, such that xn + yn = an if n is a whole number greater than 2.

  Diophantus’ problem has an infinity of solutions; specimens are x = S, y = 4, a = 5; x = 5, y = 12, a = 13. Fermat himself gave a proof by his method of infinite descent for the impossibility of x4 + Y4 = a4. Since his day xn + yn = an has been proved impossible in whole numbers (or fractions) for a great many numbers n (up to all primes* less than n = 14000 if none of the numbers x, y, a is divisible by n), but this is not what is required. A proof disposing of all n’s greater than 2 is demanded. Fermat said he possessed a “marvellous” proof.

  After all that has been said, is it likely that he had deceived himself? It may be left up to the reader. One great arithmetician, Gauss, voted against Fermat. However, the fox who could not get at the grapes declared they were sour. Others have voted for him. Fermat was a mathematician of the first rank, a man of unimpeachable honesty, and an arithmetician without a superior in history.III

  * * *

  I. This statement is sufficiently accurate for the present account. Actually, the values of the variables (coordinates and velocities) which make the function in question stationary (neither increasing nor decreasing, roughly) are those required. An extremum is stationary; but a stationary is not necessarily an extremum.

  II. The reader can easily see that it suffices to dispose of the case where n is an odd prime, since, in algebra, uab = (ua)b, where u, a, b are any numbers.

  III. In 1908 the late Professor Paul Wolfskehl (German) left 100,000 marks to be awarded to the first person giving a complete proof of Fermat’s Last Theorem. The inflation after the World War reduced this prize to a fraction of a cent, which is what the mercenary will now get for a proof.
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  CHAPTER FIVE

  “Greatness and Misery of Man”

  PASCAL

  We see . . . that the theory of probabilities is at bottom only common sense reduced to calculation; it makes us appreciate with exactitude what reasonable minds feel by a sort of instinct, often without being able to account for it. . . . It is remarkable that [this] science, which originated in the consideration of games of chance, should have become the most important object of human knowledge.—P. S. LAPLACE

  YOUNGER BY TWENTY SEVEN YEARS than his great contemporary Descartes, Blaise Pascal was born at Clermont, Auvergne, France, on June 19, 1623, and outlived Descartes by twelve years. His father Êtienne Pascal, president of the court of aids at Clermont, was a man of culture and had some claim to intellectual distinction in his own times; his mother, Antoinette Bégone, died when her son was four. Pascal had two beautiful and talented sisters, Gilberte, who became Madame Périer, and Jacqueline, both of whom, the latter especially, played important parts in his life.

  Blaise Pascal is best known to the general reader for his two literary classics, the Pensées and the Lettres écrites par Louis de Montalte à un provincial de ses amis commonly referred to as the “Provincial Letters,” and it is customary to condense his mathematical career to a few paragraphs in the display of his religious prodigies. Here our point of view must necessarily be somewhat oblique, and we shall consider Pascal primarily as a highly gifted mathematician who let his masochistic proclivities for self-torturing and profitless speculations on the sectarian controversies of his day degrade him to what would now be called a religious neurotic.

  On the mathematical side Pascal is perhaps the greatest might-have-been in history. He had the misfortune to precede Newton by only a few years and to be a contemporary of Descartes and Fermat, both more stable men than himself. His most novel work, the creation of the mathematical theory of probability, was shared with Fermat, who could easily have done it alone. In geometry, for which he is famous as a sort of infant prodigy, the creative idea was supplied by a man—Desargues—of much lesser celebrity.