768 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxvi 



Ph. Furtwangler 235 proved, in extension of Kummer's 76 work, that if 

 a l +f3 l +y l = Q, where a, /3, 7 are numbers, prime to L = (f 1), of the field 

 yfc(f), f*=l, and if a = a, /3=6, 7 = (mod L), where a, 6, c are rational, and 

 if k(t) contains no ideal belonging* to the exponent 2/4-1 modulo L, then, 

 if a;, ?/ are any two of a, b, c, 



tog (+)! _. , ,,, 



- (mocU) - 



By Mirimanoff, 180 this congruence can not hold when j = 1, 2, 3 or 4. Hence 

 if &(") does not contain ideals belonging to each of the exponents 3, 5, 7, 11, 

 Fermat's equation is impossible in numbers prime to I in &(f). The same 

 conclusion holds if the class number H is at most divisible by P. 



E. Hecke 236 proved that x l -\-y l +z l = Q is impossible in integers x, y, z, 

 each not divisible by the odd prime I, if the first factor hi of the class number 

 H of the field defined by an lib. root of unity is divisible by I, but not by P. 



D. Mirimanoff, 237 making use of his 180 criterion, proved that if 

 x p -\-y p +z p = Q has solutions prime to p, each of the six ratios x/y, is a 

 root t of 



m 1 m 1 n i ( \ 



-O (modp), fi.. = ?;- ( "J , 



p 



where ai, , a m -\ are the roots =1= 1 of z m = 1. For m = 2 or 3, at least two 

 of the six ratios are incongruent, so that our congruence, being of degree 

 <2, is an identity; taking t= I and applying 



, , 

 q(m) = 



we get q(m) = 0. Besides Wieferich's g(2) =0, we have g(3) =0. Thus the 

 initial equation is impossible in integers prime to p for all prime exponents 

 p such that either q(2) or g(3) is not divisible by p; in particular, for all 

 prime exponents of the form 2 a 3 6 l or d=2 a 3 6 . 



G. Frobenius 238 proved the last two criteria and deduced (13) from (8) 

 more simply than had Mirimanoff. 180 Set 6 2n = (-l) n -lB B , 6 2 " +1 = 0, 

 6 1 = |, so that the Bernoullian numbers are given symbolically by 



Set 



F(x, 2/)^ = z (x-iy+(x-l)>G(x, y}, 



mxG m (x] =G(x, mb)G(0, nib} , 



x 1 



mF(x)=F(x, mfy- [F(0, mb}- 



*An ideal Q, prime to L = (f 1), is said to belong to the exponent n modulo L if Q l is a 

 principal ideal () such that K=T\ (mod L n ), while there exists no unit T\ in the field 

 such that 7jK = r 2 (mod L" 4 " 1 ), where r\ and r 2 are rational numbers. 



235 Gottingen Nachrichten, 1910, 554-562. 



236 Ibid., 420-4. 



237 Comptes Rendus Paris, 150, 1910, 204-6. Reproduced. 246 



238 Sitzungaber. Akad. Wiss. Berlin, 1910, 200-8. 



