Men of Mathematics (14 page)

The next point is of great importance in the life of Fermat, also in the history of mathematics. Consider the numbers 3, 5, 17, 257,
65537.
They all belong to one “sequence” of a specific kind, because they are all generated (from 1 and 2) by the same simple process, which will be seen from

3
= 2 + 1, 5 = 2
²
+ 1, 17 = 2
⁴
+ 1, 257 = 2
⁸
+ 1,
65537
= 2
¹⁶
+ 1;

and if we care to verify the calculation we easily see that the two large numbers displayed above are 2
³²
+ 1 and 2
⁶⁴
+ 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), 2
¹
, 2
²
, 2
³
,
2
⁴
, 2
⁶
, 2
⁶
. Namely, our sequence is 2
²
” 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” 2
^2
n
” + 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 2
²ⁿ
+ 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
n
^p
—n
is divisible by
p.
For example, taking
p
= 3,
n
= 5, we get 5
³
– 5, or 125 – 5, which is 120 and is 3 × 40; for
n
= 2,
p
= 11, we get 2
¹¹
– 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
n
^p
—
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 methods 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 4
n
+ 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 4
n
−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, = l
²
+ 6
²
, and that there are no other squares
x
²
and
y
²
such that 37 =
x
²
+jy
²
. For the prime 101 we have l
²
+ 10
²
; for 41 we find 4
²
+ 5
²
. 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 4
n
+ 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 [4
n
+ 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
4
n
+
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 = 3
³
and 25 = 5
²
. Thus we observe that
y
^z
= x
²
+ 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.