CHAP, xxvi] FERMAT'S LAST THEOREM. 747 



2m+l 



summed for r = 1, 2, . The first formula had been given earlier. 88 

 Housel 89 proved Catalan's 38 empirical theorem that two consecutive 



integers, other than 8 and 9, can not be exact powers [with exponents >!]. 

 E. Catalan 90 stated this theorem and those given under Catalan. 1220 

 Catalan 91 set p=x+y+z, P = p n x n y n z n and proved that the 



quotient Q of P by (x+y)(y+z)(z+x) is (for n odd >3) 



where Hi = p, H^Zx^+Zxy, H 3 = 2x 3 +2x 2 y+xyz, 



H q (x, z) =x*+zx- 1 +z 2 x- 2 -{ 

 If n is a prime the coefficients of P and Q are divisible by n. Also, 



where is a polynomial in x, y, z with integral coefficients. 



G. C. Gerono 92 proved that, if x or y is a prime, x m = y n +l holds in 

 positive integers >1 only when x = n = 3, y = m = 2. See Carmichael. 226 



A. Genocchi 93 stated that x 4 -\-6x 2 y 2 y 4 /7=z 2 is impossible in integers. 

 Hence x 7 +y 7 -\-z 7 = Q is not satisfied by values of x, y, z which are roots of a 

 cubic equation with rational coefficients, a generalization of Lamp's 28 

 theorem. 



E. Laporte 94 would deduce Fermat's last theorem from the fact that the 

 series of powers higher than the second are formed by the summation of 

 terms of arithmetical progressions preceded by extraneous terms. 



Moret-Blanc 95 proved that the only positive integral solutions of 



are y = Q; y = l, z = 2; y = 2, x = 3. A. J. F. Meyl 96 showed that the only 

 positive integral solutions of (x+l) v =x y+l -{-l are z = 0, x = y = l, x = y = 2. 



88 N. M. Ferrers and J. S. Jackson, Solutions of the Cambridge Senate-House Problems for 



1848-1851, pp. 83-85. 



89 Catalan's Melanges Math., Liege, ed. 1, 1868, 42-48, 348-9. 



90 Ibid., 40-1; Revue de ^instruction publique en Belgique, 17, 1870, 137; Nouv. Corresp. 



Math., 3, 1877, 434. Proofs by Soons. 172 



91 Melanges Math., ed. 1, 1868, No. 47, 196-202; Mem. Soc. Sc. Liege, (2), 12, 1885, 179-185, 



403. (Cited in Bull, des sc. math, astr., (2), 6, I, 1882, 224.) 



92 Nouv. Ann. Math., (2), 9, 1870, 469-471; 10, 1871, 204-6. 



93 Comptes Rendus Paris, 78, 1874, 435. Proof, 82, 1876, 910-3. 



94 Petit essai sur quelques methodes probables de Fermat, Bordeaux, 1874. Reprinted in 



Sphinx-Oedipe, 4, 1909, 49-70. 



95 Nouv. Ann. Math., (2), 15, 1876, 44-6. 

 98 Ibid., 545-7. 



