148 REPORT — 1860. 



x = 0, mod»; also, to adapt the formulae of art. 30 to our present par* 

 x— 1 



pose, let _A '=a, m=\', n=k\'; it will result from these substitutions, that 

 »/* (m _a ) = 0, mod/?, if A and A satisfy the inequality [A] + [AA] >X, where 

 [A] and [AA] are positive numbers less than X, and congruous, mod X, to A 

 and AA respectively. If we represent by /(a) the ideal prime factor of p 

 which appertains to the substitution a=w, this may be expressed by saying 



that ^i(a) contains the factor/(a-'<), if - + | - >X, the symbols - 



and - denoting the least positive numbers satisfying the congruences 



hx = l, mod X, and hx = k, mod X. Assigning, therefore, to the number h 

 every positive value less than X compatible with this condition, we may write 



Ma)=±a«n/(a-*), 

 + a* being a simple unit which may be determined by the congruence 

 \p k (a) = — 1, mod (1 —a) 2 * : it is not necessary to add a real complex unit, 

 for a reason which has already appeared (see art. 56, supra). From the 

 expression for \p k (a) a still simpler formula for F (a, x) K may be obtained, 

 viz. m—\— 1 [11 



[F (a, x)y= ±a. s n [/(«-»)] L '" J t- 



»i = l 



61. Application to the Last. Theorem of Fermat. — The second investigation 



to which we shall advert relates to the celebrated proposition known as the 



" Last Theorem of Fermat," viz. that the equation x n +y n =z n is irresoluble, 



in integral numbers, for all values of n greater than 2|. As Fermat himself 



* The numbers ^ k (a) are primary according to M. Kururner's definition (art. 52) ; for 



F ( a x)¥ (oft X~) 



^ k (a) = ' k+ \ — — i = 2«yi+*i's, the summation extending to every pair of values of 



y l and y„ that satisfy the congruence y*«+y*fc=l, mod^, in which y represents the same 

 primitive root of p that occurs in the expression F («, x). llcnce ■4' k (l)=p—2 = —I, 

 mod X, and -4> k («) $ k (« _1 )= d p=l = [^ t (l)] a , mod \. Also ^ («)-^ t (1) is divisible 

 by (l-«) 2 ; for^' A (l)=sfjr 1 +*y i )=J(l+*) Q»-l) Q>-2), observing that y : and y 2 

 each receive all the values 1, 2, ...^ — 2 in succession. We have, therefore, the con- 

 gruence ^'j (1) =0, mod X, from which it follows (see a note on the next article) that 

 •^ (a.) es^j. (1), mod (1 — a) 2 , or ^ A («)=— 1, mod (1 — a) 2 , as in the text. 



t Liouville> vol. xvi. p. 448. M. Kummer has also extended his solution of this problem 

 to the case in which n is any divisor of p— 1. See the memoir quoted in the last article, 

 sect. 11. 



J Fermat's enunciation of this celebrated theorem is contained in the first of the MS. notes 

 placed by him on the margin of his copy of Bachet's edition of Diophantus. It would seem 

 that this copy is now lost ; but in the year 16/0 an edition of Bachet's Diophantus was pub- 

 lished at Toulouse, by Samuel de Fermat (the son of the great geometer), in which these 

 notes are preserved (Diophanti Alexandrini Arithmeticorurn libri sex, et de Numeris Mult- 

 angulis liber unus, cum commentarris C. G. Bacheti V. C. et observationibus D. P. de Fermat 

 senatoris Tolosani. Tolosae 1670). The theorems contained in them are, with a few excep- 

 tions, enunciated without proof; and it may be inferred from the preface of S. Fermat, that 

 he found no demonstration of thein among his father's papers. Nevertheless, in the case of 

 several of these propositions, we have the assertion of Fermat himself, that he was in posses- 

 sion of their demonstration ; and although, when we consider the imperfect state of analysis 

 in his ti:ne, it is surprising that he should have succeeded in creating methods which sub- 

 sequent mathematicians have failed to rediscoyer, yet there is no ground for the suspicion 

 that he was guilty of an untruth, or that he mistook an apparent for a real proof. In fact 

 these suspicions are refuted, not only by the reputation for honour and veracity which he 

 enjoyed among his contemporaries, and by the evidence of singular clearness of insight 

 which his extant writings supply, but also by the facts of the case itself. It would be iuex- 



