CHAP, xxvi] FERMAT'S LAST THEOREM. 737 



proofs, it is shown 26 that if 



and a6 have a common factor, it divides the last term =bn(afr) (n ~ 1)/2 , and 

 hence is the prime n if a and b are relatively prime. Since the coefficients 

 n, w(w 3)/2, are divisible by n, the exponent of the highest power of n 

 dividing a n b n exceeds by unity that in ab. 



F. Paulet 27 attempted to prove Fermat's last theorem, but concluded 

 without proof that o: = /3 in acr = j3s, where 



a = bmx z (p q)a, (3 = ar+(pq)c+s. 



In his second proof he equated corresponding summands of equal sums. 



G. Lame" 28 proved' that x'-\-y 7 -\-z ! = Q is impossible in relatively prime 

 integers. One of the unknowns, say x, is divisible by 7 (Legendre 17 ). 

 It is shown that x+y+z = 7AP, P = /j.vp } where ju, v, p, 7 are relatively 

 prune integers such that 



He made use of the lemma (pp. 197-8) that [Bouniakowsky 34 ] 



Thus A must be a square B 2 . Then 

 Za = 27 2 P, Za 2 + Sa& = BD, abc = 7 6 P 7 , 32a 4 + 10Za 2 6 2 = 2 4 14 . 



Eliminatmg a, b, c, we get an equation whose solution is shown to depend 

 upon the impossible equation 



C7 8 -3 -7 4 C7 4 7 4 +2 4 7 5 7 8 = TF 4 . 



For simplifications of this proof, see Lebesgue 30 and Genocchi. 85 



A. Cauchy 29 reported on Lame's preceding paper and stated that his 



lemma is obtained by taking n = 7 in the generalization that (x+y} n x n y n 



is algebraically divisible not only by nxy(x+y) but also (if n>3) by 



q=x 2 -\-xy-}-y 2 , and if n = 6&+l by g 2 . 



V. A. Lebesgue 30 simplified Lame's 28 proof by use of the lemma that 



is impossible in odd integers p, q, r, relatively prime in pairs, r 4=0, if a is 

 a positive integer. 



Lebesgue 31 proved that if X n +Y n = Z n is impossible in integers, then 

 x Zn -\-y 2n = z 2 is impossible. 



28 Also in Nouv. Ann. Math., 7, 1848, 239, 307-8. 



"Corresp. Math, (ed., A. Quetelet), 11, 1839, 307-313. 



28 Comptes Rendus Paris, 9, 1839, 45-6; Jour, de Math., 5, 1840, 195-211. M6m. pr6sent6s 



divers savants Acad. Sc. de 1'Institut de France, 8, 1843, 421-437. 

 "Comptes Rendus Paris, 9, 1839, 359-363; Jour, de Math., 5, 1840, 211-5. Oeuvres de 



Cauchy, (1), IV, 499-504. 



30 Jour, de Math., 5, 1840, 276-9, 348-9 (removal of obscurity in proof of lemma). 



31 Ibid., 184-5. 



48 



