756 



HlSTOKY OF THE THEORY OF NUMBERS. 



[CHAP. XXVI 



and m not divisible by 5 (since x 5 +y 5 =zl is impossible by Legendre 18 ). 

 By the residues of z 5 , x 5 -}-y 5 modulo 25, we see that m is not in the set 

 2, 4, 7, 9, 12, -, 2n+[(n-l)/2]. For z divisible by 5, we have z = 5tr, 

 x+y = 2 m 5*fi. Proceeding as did Legendre, we find that the equation is 

 impossible. 



D. Mirimanoff 146 proved by use of ideals that z 37 +2/ 37 +2 37 = is impossible 

 in integers. 



H. Dutordoir 147 expressed his belief that a n +b n = c n is impossible in 

 integers if n is a rational number other than 1 and 2. The fact that it is 

 impossible when n = 1/2 and one of a, 6, c is not a perfect square is a case of 

 the impossibility of 



when c is different from a and b, and one of the four numbers a, , d is 

 not a square (Euclid, Elements, X, 42). 



A. S. Bang 148 pointed out errors in various elementary proofs of special 

 cases of Fermat's last theorem. 



G. Korneck 149 claimed to prove Fermat's last theorem by means of the 

 Lemma : If n and k are relatively prime (n odd) and divisible by no square 

 >1, then in every solution in integers of nx*+ky z = z n , x is divisible by n. 

 E. Picard and H. Poincare 150 pointed out the falsity of this Lemma by 

 citing the examples n = 3, k = l, x = y = z = 4, and n = 5, k = 3, x = l, 2/ = 3, 

 z = 2. The Jahrbuch Fortschritte der Math., 25, 1893, 296, pointed out 

 that 3 of Korneck's paper shows a lack of knowledge of the nature of 

 algebraic numbers. 



Malvy 151 noted that, if a is a primitive root of a prune p = 2 n -\-l, and if 

 in o 2M+1 +l=o* (mod p) we give to p. the values 1, 2, , 2"" 1 , we obtain for 

 h as many even as odd values. If in a 4M+2 +l=a* we give to /* the values 

 1, -, 2 n ~ 2 , we obtain a even and /? odd values for h, while if p = 17, a = 3 

 or p = 257, a = 5, we have a = /3. 



E. Wendt 152 proved that if n and p = mn+l are odd primes, 



r n +s n +t n = Q (mod p} 



has only solutions in which r, s or t is divisible by p if and only if p is not a 

 divisor of 



146 Jour, fur Math., Ill, 1893, 26-30. 



147 Ann. Soc. Sc. Bruxelles, 17, 1, 1893, 81. Cf. Maillet. 286 



148 Nyt Tidsskrift for Math., 4, 1893, 105-7. 



149 Archiv Math. Phys., (2), 13, 1894 (1895) ; 1-9. He noted, pp. 263-7, that the Lemma fails 



for n = 3, A; = 1, and so gave a separate proof of the impossibility of x 3 +y 3 =z 3 . 

 160 Comptes Rendus Paris, 118, 1894, 841. 



151 L'interm6diaire des math., 1, 1894, 152; 7, 1900, 193 (repeated). 

 162 Jour, fur Math., 113, 1894, 335-347. 



