FERMAT'S LAST THEOREM, a statement which is famous in the history of mathematics, namely that there do not exist integers x, y and z, none of which being zero, which satisfy x"-+-y'Z=z" (I) a being a given integer > 2. It was first given by Fermat who wrote, about the year 1637, upon the margin of his copy of the works of Diophantus, "I have discovered a truly remarkable proof which this margin is too small to contain." He did not publish his proof, however, and no complete demonstration has as yet been discovered.
The theorem was proved by Euler for n = 3 and 4. To prove it in general, it is then not difficult to see that it is sufficient to demonstrate the impossibility of xl+yld-zl=o (2) in non-zero integers x, y and z for any odd prime l > 3. This was proved by Legendre in 1823 for l= 5, and later by Lebesgue for 1= 7. These proofs excited great interest among the mathema ticiaris of the time and many efforts were made to extend them to other values of 1, a number of them resulting in errors. For example, the celebrated mathematicians Lame and Cauchy pub lished many results concerning (2) which were based on assump tions later proved false. The discussion and study of these mis takes led to the formulation, by Kummer, of one of the most powerful and fruitful concepts which has ever been introduced into mathematics, i.e., the notion of ideal numbers. By means of this idea Kummer was able to prove (in 185o) that (2) is impossible in integers x, y and z, none zero, for all primes 1 for which none of the numerators of the Bernoulli numbers (q.v.) a= I, 2, . . . , (l-30, is divisible by I, where B, =1/6, etc. Primes 1 of this character are called regular. By extensive numerical computations he found that the only primes less than zoo which are not regular are 37, S9 and 67, and in a later article published in 1874 he obtained results from which it may be inferred that the only primes which are not regular be tween the limits ioo and 166 are i o i , 103, 131, 149 and 157. In the year 1857 Kummer published another memoir in which he concluded after a long and complicated procedure that (2) is impossible under three assumptions concerning the nature of the algebraic field defined by where = He finds, again by the use of extensive numerical computations, that these three assumptions are satisfied for 1=37, 59 and 67, and hence that (2) is impossible for all primes l< 1 oo.
In papers published recently by H. S. Vandiver several errors were pointed out in the arguments employed by Kummer in his paper of 1857 and the necessary corrections were made. The Tat ter's extensive numerical computations concerning the regular and non-regular primes 1 which are less than 1 oo and which we have already referred to, have not all been checked, however, by other writers. If we assume they are all correct then (2) has been proved impossible for all primes l< 1 oo. For the statement of later results we shall divide the discussion into two cases. If in (2), x, y and z are prime to each other and to 1, this condition will be referred to as case I. of Fermat's last theorem; if x, y and z are prime to each other and one of them is divisible by 1, the condition will be called case II. of the theorem. We shall also confine ourselves to mentioning only those criteria for the solution of (2) which do not involve in their enunciation the theory of algebraic numbers.
Legendre published in 1823 the following theorem due to Sophie Germain :—If there exists an odd prime p such that the congruence has no set of integral solutions u, v and w, each not divisible by p, and such that 1 is not the residue of the lth power of any integer modulo p, then (2) has no solutions x, y and z, each prime to 1. By means of this result Sophie Germain proved that (2) is impossible in case I. for all primes /< 1 oo.
In 1857 Kummer proved that if (2) is satisfied in case I. then r = I, 2, • • • , (l--3)/2, where is, as before, the nth Bernoulli number and the other symbol designates the result obtained by taking the (1— 2 fa) th derivative with respect to v of the logarithm of x+evy, where e is the Napierian base, and setting v=o. In 1905, Mirimanoff showed that the criteria (4), with the others obtained from them by replacing x by z, etc., are equivalent to for the solution of (2) in case I. The criteria (4a) and (4b) are now generally referred to as the Kummer criteria.
In 1908 Dickson, using Sophie Germain's theorem, proved that (2) is impossible in case I. for all primes l< 7,000. Wieferich, in 1909, by the use of (4a) and (4b) proved that, if (2) is possible in case I., then (mod and in 1910 Mirimanoff ob tained, on the same assumption, (mod Furtwangler in 1912 proved that, if (2) is satisfied in case I. and r is a factor of x or of where =o (mod 1), then (mod This result includes those of Wieferich and Mirimanoff just re ferred to. In the year 1925 Beeger found that the only primes l<14,000, for which (mod are 1=1,093 and 1=3,511, the case 1=1,093 having been previously noted by Meissner. Using these results with those of Dickson he concludes that (2) is im possible in integers x, y and z prime to 1, for all primes l< 14,00o. H. S. Vandiver obtained in 1926 the theorem that, if there exists an odd prime p= I +ml, such that m< To/ and (mod p), has no set of integral solutions u, v, w, each not divisible by p, then (2) has no solution in case I. He also proved (192 5) that if (2) is satisfied in case I. then i= I, 2, ... , 1-1; t having the same meaning as in (4a).
As to case II. of the theorem, i.e., when one of the integers x, y and z in (2) is divisible by 1, no result has been published which is known to represent an advance over Kummer's results in his memoir of 11857. In general, as to the present attitude towards Fermat's last theorem, there is evident among mathe maticians a growing opinion that it is not true. In particular, several specialists in the theory of numbers think that it is quite possible that there exist odd prime integers l and integers x, y and z prime to each other, z=o (mod 1) such that However, in case I., everything indicates that it is true, although it has not been proved. That it has not been proved for this case is one of the most amazing facts in present-day mathematics, and for this reason alone the great celebrity of the theorem is perhaps justified. Also the many attempts to solve it by competent mathe maticians have led to a number of remarkable developments in number theory, including very abstract conceptions and profound results, arrived at only after long chains of reasoning. In 1907 the Wolfskehl prize of 1 oo,000 marks, for the first demonstration of the theorem was established. This led to the publication of sev eral thousand erroneous "proofs" by individuals who were, for the most part, not at all equipped mathematically to cope with the problem.