42 ARITHMETICAL RECREATIONS [CH. II 



definitely that he had a valid proof — demonstratio mirabilis 

 sane — and the fact that no theorem on the subject which he 

 stated he had proved has been subsequently shown to be false 

 must weigh strongly in his favour; the more so because in 

 making the one incorrect statement in his writings (namely, 

 that about binary powers) he added that he could not obiain 

 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. The subject is in itself one of peculiar 

 interest and elegance, but its conclusions have little practical 

 importance, and for long it was studied by only a few mathe- 

 maticians. 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 re = 5; 

 in 1832 Lejeune Dirichletf gave one for n = 14; and in 1840 

 Lame and Lebesgue J gave proofs for n = 7. 



The proposition appears to be true, and in 1849 Kummer§, 

 by means of ideal primes, proved it to be so for all numbers 

 which satisfy certain Bernoullian conditions. The only numbers 

 less than 100 which do not do so are 37, 59, 67, and the theorem 

 can, by other arguments, be proved for these three cases. Other 

 tests have been established; for instance, A. Wieferich|| has 

 shown that if the equation is soluble in integers prime to n, 

 where n is an odd prime, then 2 n_1 — 1 is divisible by rf. No 

 exception to the theorem has yet been found, and by one means 

 or another, the number of cases which require special discussion 

 has been reduced to but few. I may add that to prove the 

 truth of the proposition when n is greater than 4, obviously it 



* Reprinted in his ThSorie des Nombres, Paris, 1830, vol. n, pp. 361 — 368: 

 see also pp. 5, 6. 



t Crelle's Journal, 1832, vol. ix, pp. 390—393. 



X Liouville's Journal, 1841, vol. v, pp. 195—215, 276—279, 348—349. 



§ References to Rummer's Memoirs are given in Smith's Report to the 

 British Association on the Theory of Numbers, London, 1860. 



|| Crelle's Journal, Berlin 1909, vol. cxxxvi, pp. 293—302. 



