738 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. XXVI 



J. Liouville 32 noted that if u n +v n = w n is impossible in integers not zero, 

 then z 2n y 2n = 2x n is impossible. 



Cauchy 33 expressed (x+y) n x n y n in terms of x 2 +xy+y 2 and xy(x+y) 

 for n odd ^13. 



V. Bouniakowsky 34 proved for m = 2, 3, 4, 5, 6, 7 that 



is impossible if R is rational and the radicals irrational. For m = 7 set 

 C=(AB) 1 ' 7 . We get R 7 -A-B = 7RC(R 2 -C) 2 , which implies the lemma 

 of Lam6 28 (Cauchy 29 ). For, by setting A = a 7 , B = b 7 , R = a+b, C = ab, we get 



E. E. Kummer 35 submitted to Dirichlet about 1843 the manuscript 

 giving what he then believed to be a complete proof of Format's last theorem. 

 Dirichlet declared that the proof would be correct if it were shown not 

 only that every number a +!-}- +a x _ia x ~ (where a is a primitive 

 Xth root of unity and the a's are ordinary integers) is always a product of 

 indecomposable numbers of that form, as shown by Kummer, but also 

 that this were possible in only one way, which is unfortunately apparently 

 not the case. 



Frizon 36 announced a uniform process applicable to prime exponents 

 ^31. 



V. A. Lebesgue 37 supplemented Dirichlet 's 20 results by proving that, 

 if A has no prime factor 10w+l and no factor which is a fifth power, 

 x 5 +y b = AB 5 u 5 is impossible in integers if A is a multiple of 5, or if A = 2, 

 3, 4, 6, 8, 9, 11, or 12 (mod 25). A like treatment is 

 apparently not applicable to the remaining cases J. = l, 7 (mod 25). 

 The equation x w y w = Az 5 is impossible if A has no prime factor 10w+l. 

 As auxiliary propositions, a 2 = 6 4 +506 2 c 2 +125c 4 is impossible, while 



a 2 = 6 4 +106 2 c 2 +5c 4 , 



which can be reduced by descent to the case in which b and c are odd, is 

 impossible if = 5-2*' -h 2 . 



E. Catalan 38 expressed his belief that x m y n = 1 holds only for 3 2 2 3 = 1. 



S. M. Drach 39 argued that x n +y n = z n is impossible in integers if 

 n = 2m+l > 1. For, by Euler's Algebra, 2, Ch. 12, 



Y = c m q n +2A i q n - 2i p 2i c m - i a i , Z = a m p n +^A i p n - 2i q 2i a m ~ i c i 

 satisfy aZ 2 -cY 2 = (ap 2 - cq 2 ) n if A < = ( 2 " ) . Take a = z, Z = z m , c = y, Y=y m . 



32 Jour, de Math., 5, 1840, 360. 



33 Exercices d'analyse et de phys. math., 2, 1841, 137-144; Oeuvres, (2), XII, 157-166. 



34 M&n. Acad. Sc. St. PStersbourg (Math.), (6), 2, 1841, 471^92. Extract in Bull. St. 



Peters., VIII, 1-2. 

 36 K. Hensel, Gedachtnisrede auf E. E. Kummer, Abh. Gesch. Math. Wiss., 29, 1910, 22. 



CCf. the less technical address by Hensel, E. E. Kummer und der grosse Fermatsche 



Satz, Marburger Akademische Reden, 1910, No. 23.] 

 88 Comptes Rendus Paris, 16, 1843, 501-2. 



17 Jour, de Math., 8, 1843, 49-70. 



38 Jour, fiir Math., 27, 1844, 192. Nouv. Ann. Math., 1, 1842, 520; (2), 7, 1868, 240 (re- 

 peated by E. Lionnet). For n = 2, Lebesgue 68 of Ch. VI. 



18 London, Edinburgh, Dublin Phil. Mag., 27, 1845, 286-9. 



