Fermat seems to have discovered its truth first for the case n = 3, and then for the case n = 4. His proof for the former of these cases is lost, but that for the latter is extantk , and a similar proof for the case of n = 3 was given by Euler . These proofs depend upon showing that, if three integral values of x, y, z can be found which satisfy the equation, then it will be possible to find three other and smaller integers which also satisfy it: in this way finally we show that the equation must be satisfied by three values which obviously do not satisfy it. Thus no integral solution is possible. It would seem that this method is inapplicable except when n = 3 and n = 4.
Fermat’s discovery of the general theorem was made later. An easy demonstration can be given on the assumption that every number can be resolved into prime (complex) factors in one and only one way. That assumption has been made by some writers, but it is not universally true. It is possible that Fermat made some such supposition, though it is perhaps more probable that he discovered a rigorous demonstration. At any rate he asserts definitely that he had a valid proof— demonstratio mirabilis sane—and the fact that every other theorem on the subject which he stated he had proved has been subsequently veri- fied must weigh strongly in his favour; especially as in making the one statement in his writings which is not correct he was scrupulously careful to add that he could not obtain a satisfactory demonstration of it.
It must be remembered that Fermat was a mathematician of quite the first rank who had made a special study of the theory of numbers. That subject is in itself one of peculiar interest and elegance, but its conclusions have little practical importance, and since his time it has been discussed by only a few mathematicians, while even of them not many have made it their chief study. This is the explanation of the fact that it took more than a century before some of the simpler results which Fermat had enunciated were proved, and thus it is not surprising that a proof of the theorem which he succeeded in establishing only towards the close of his life should involve great difficulties.
In 1823 Legendre obtained a proof for the case of n = 5; in 1832 Lejeune Dirichlet gave one for n = 14, and in 1840 Lam´e and Lebesgue gave proofs for n = 7.
The proposition appears to be true universally, and in 1849 Kummer , by means of ideal primes, proved it to be so for all numbers except those (if any) which satisfy three conditions. It is not known whether any number can be found to satisfy these conditions, but it seems unlikely, and it has been shown that there is no number less than 100 which does so. The proof is complicated and difficult, and there can be little doubt is based on considerations unknown to Fermat. I may add that to prove the truth of the proposition when n is greater than 4, it obviously is sufficient to confine ourselves to cases where n is a prime, and the first step in Kummer’s demonstration is to show that in such cases one of the numbers x, y, z must be divisible by n.
Naturally there has been much speculation as to how Fermat arrived at the result. The modern treatment of higher arithmetic is founded on the special notation and processes introduced by Gauss, who pointed out that the theory of discrete magnitude is essentially different from that of continuous magnitude, but until the end of the last century the theory of numbers was treated as a branch of algebra, and such proofs by Fermat as are extant involve nothing more than elementary geometry and algebra, and indeed some of his arguments do not involve any symbols. This has led some writers to think that Fermat used none but elementary algebraic methods. This may be so, but the following remark, which I believe is not generally known, rather points to the opposite conclusion. He had proposed, as a problem to the English mathematicians, to show that there was only one integral solution of the equation x 2 + 2 = y 3 : the solution evidently being x = 5, y = 3. On this he has a note to the effect that there was no difficulty in finding a solution in rational fractions, but that he had discovered an entirely new method—sane pulcherrima et subtilissima—which enabled him to solve such questions in integers. It was his intention to write a work† on his researches in the theory of numbers, but it was never completed, and we know but little of his methods of analysis. I venture however to add my private suspicion that continued fractions played a not unimportant part in his researches, and as strengthening this conjecture I may note that some of his more recondite results—such as the theorem that a prime of the form 4n+ 1 is expressible as the sum of two squares—may be established with comparative ease by properties of such fractions.