CHAP, xxvi] FERMAT'S LAST THEOREM. 761 



we conclude that for each power we get six representations of a prime by 

 (u, v); but only three representations of 5. A composite number has 2 P 

 representations if p is the number of its distinct prime factors 5nl. 

 Take Zi = (a, 6). We get u, v such that z\ = (u, v) by using 



(a, 

 Then 



(10) (x-yY = v 



The product of the square root of the last sum by (s, t) gives Az 5 , so that 

 we have the general form of A. Taking x+y arbitrary, we get x y and 

 thens, tby (10). 



Mirimanoff 180 considered 



(11) z x +2/ x +z x = 



for the case in which no one of the integral solutions x, y, z is divisible 

 by the odd prime X. By use of Kummer's congruences (8), he proved 

 that (11) is impossible in integers prime to X if at least one of the Bernoul- 

 lian numbers* B^i, B^, B^ 3) B^ is not divisible by X, where 



also, for every X<257. In terms of Kummer's P t -()=jPi(l, J), he defined 

 the polynomials 



(12) ^0 



modulo X. Thus Kummer's criterion (8) is equivalent to the following. 

 If (11) has solutions prime to X, each of the six ratios t = x/y, , zfx satisfies 

 the congruences 



(13) 0x_i(0H=0, J5 (x _,-,/20i(0=0 (mod X) (i = 3, 5, , X-2). 



An equivalent criterion not involving Bernoullian numbers is that each 

 of the six ratios satisfies the congruences 



(14) x -i(0=0, x -.-(0*i(0=0 (mod X) (i = 2, 3, -, v). 



E. Maillet 181 proved by Kummer's methods that x a -\-y a = az a (a>2) has 

 no real integral solutions =j= if a is divisible by 4 ; or if a is even and divisible 

 by a prime 4n+3; or if 2<a^lOO, a + 37, 59, 67, 74; or if a has no prime 

 factor >17. Likewise for x a +y a = baz a if a is divisible by 4 and 6 is not; 

 or if a is of the form 4n+2 and has a prime factor X = 4/i+3 such that 6 is 

 not divisible by X*" 1 ; or if a = p i j b<p,p being a prime =s5 not exceptional in 

 the sense of Kummer; or if o = 3*', & = 2 or 4, i^2. Probably the second 

 equation is impossible in integers 4= if b = 1 or 2, a > 2 or a > 3, respectively. 



R. Sauer 182 proved that x n = y n +z n , n>2, does not hold if x or y or z 

 is a power of a prune. 



U. Bini 183 noted that, if x+y+z = and fc = 2m+l, s=x k +y k +z k is 

 divisible by xyz. If l/x+lfy+l/z = Q and & = 3/i+2, s is divisible by 



180 Jour, fur Math., 128, 1905, 45-68. 



* If B v -\ or B,,-2 is not divisible by X, the conclusion was drawn by Kummer. 76 



181 Annali di mat., (3), 12, 1906, 145-178. Abstracts in Comptes Rendus Paris, 140, 1905, 



1229; Me"m. Acad. Sc. Inscr. Toulouse, (10), 5, 1905, 132-3. 



182 Eine polynomische Verallgemeinerung des Fermatschen Satzes, Diss., Giessen, 1905. 



183 Periodico di Mat., 22, 1906-7, 180-3. 



