CHAP, xxvi] FERMAT'S LAST THEOREM. 759 



number of classes of these complex integers, and hence for a value of t 

 exceeding a certain limit depending on X. He 165 later proposed the problem 

 that the last theorem be proved without the restriction that x, y, z are 

 prime to X. 



I. P. Gram's 166 paper was not available for report. 



E. Maillet 167 applied Rummer's methods to z x +?/ x = c2 x , where X is a 

 regular prime. The equation is impossible in integers if c = X. It is im- 

 possible in real relatively prime integers not divisible by X if c = A\ f , 

 s = fcX+|8^1, /8 = or 1, when A = l or r\ l - -r-*', where n, , r t are distinct 

 primes =t=X, belonging to exponents/!, ,/ modulo X such that 



1 1 ^X-3 



i- . . . _] < - 



/i TA x-i ; 



in particular, if A = rl 1 , r^l (mod X) . For r a prune and b < X, the equation 

 with c = r b is impossible in real integers if r b ^ 1-HX (mod X 2 ), where 

 t has at least one of the values 1, , X 1 ; or if X = 5, 7, 17, r 6 =4 (mod X 2 ) ; 

 or if X = ll, r 6 -5 or 47 (mod II 2 ); or if X = 13, r b =l7 (mod 13 2 ). Finally, 

 x 7 +y 7 = cz 7 is impossible in real integers for c a prime of one of the forms 

 49&3, 4, 5, 6, -8, 9, 10, -15, 16, -22, 23 or 24. 



H. J. Woodall 168 noted that x m -\-y m 1 is divisible by xy if y x m 1 

 (m even) or if x = 2, y = 2 m l (m odd). 



T. R. Bendz 169 stated that x n -\-y n = z n has integral solutions if and only 

 if a 2 = 4jS"+l has rational solutions [Euler 8 ], as follows from 



He proved Abel's 16 formulas, also x+y=z (mod 3) and (p. 30) 



(x+y) n x n y n =Q (mod n 3 ), 



when no one of x, y, z is divisible by n. 



F. Lindemann 170 attempted to prove that x n = y n -\-z n is impossible if 

 n is an odd prime. He later (p. 495) recognized the error in the computa- 

 tion, but stated that his work gives the first proof of Abel's 16 statement 

 that if x, y, z are 4= and relatively prime in pairs 



if no one of x, y, z is divisible by n, while, if z is divisible by n, 

 2x, 2y = p n +q n n n ~ 1 r n , 2z = p n q n +n n ~ 1 r n . 



If x+y+z is divisible by n x , then, in (2), a=(3=y= 1 (mod n*" 1 ). 



D. Gambioli 171 proved de Jonquieres' 117 theorems, and the fact that in 

 x n -\-y n = z n (n>l), z is composite if n has an odd factor, or if x and y are 



166 Congres intertoat. des math., 1900, Paris, 1902, 426-7. 



166 Forhandlingar Skandinaviska Naturforskare, Gotheborg, 1898, 182. 



167 Comptes Rendus Paris, 129, 1899, 198-9. Proofs in Acta Math., 24, 1901, 247-256. 



168 Math. Quest. Educ. Times, 73, 1900, 67. 



169 6fver diophantiska ekvationen x n -\-y n =z n , Diss., Upsala, 1901, 34 pp. 



170 Sitzungsber. Akad. Wiss. Miinchen (Math.), 31, 1901, 185-202. 



171 Periodico di Mat., 16, 1901, 145-192. 



