CHAP, xxvi] FERMAT'S LAST THEOREM. 757 



which is the resultant of x m = 1, (x-\-l} m = 1. For, if we multiply the con- 

 gruence by oo n , where co=l, we obtain a congruence of the form x+l = y 

 (mod p}, where x and y are nth powers, so that their rath powers are con- 

 gruent to unity. 



He proved Legendre's 17 result concerning the cases ra = 2, 4, 8, 16. If 

 m = 2"n k can be chosen so that ran+1 is a prune not dividing D m , where v 

 is not divisible by the prime n, then a n = b n +c n (n>2) is not solvable in 

 integers all prime to n. If mn-\- 1 is a prime dividing neither D m nor n m 1, 

 the same conclusion holds. [This result differs only in form from that 

 by Sophie Germain 17 ]. 



D. Hilbert 153 gave a simplification of Kummer's 63 proof of Fermat's 

 theorem for regular prime exponents, and a proof that 4 +/3 4 = 7 2 is im- 

 possible in complex integers a -\-bi. 



G. B. Mathews 154 noted that, if p is an odd prime, and x, y, z are solutions 

 of x p +y p +z p = Q, it is possible to choose k in an infinitude of ways such 

 that kp+l = q is a prime not a factor of x, y, z, or y p z p , etc., and such 

 that k is not divisible by 3. Then, since x p , y p , z p are distinct roots of t k = 1 

 (mod g), their sum is divisible by g. Let r = e ini " e and P k = H(r a +r f} +r y }, 

 where the product extends over all triples of roots r a , r , r y of x k = 1. Then 

 Pk= dbtt*, where u k is a positive integer. Thus u k = (mod q) if and only 

 if three roots of x k =l (mod q) have a sum divisible by q. Hence if it 

 could be proved that for a given p there is an infinitude of primes kp+1 

 for which u k =0 (mod q) is not satisfied, Fermat's theorem would follow 

 [Libri 24 ]. 



E. de Jonquieres 155 noted that, if n>2, it is not possible to express 

 c and b as algebraic functions of p, q such that c n b n becomes (pq) n 

 identically, and stated that this does not imply the impossibility of integral 

 solutions. 



G. Speckmann 156 discussed T x DU x = m x . 



V. Markoff 157 noted that Lucas' 138 proof of Abel's 16 theorem that 

 a n = b n -\-c n (n an odd prime) is impossible when a, 6 or c is a prime is in- 

 complete as the case a = b-\-l is not treated. He asked if (x+l) n = x n +y n 

 is impossible. 



P. Worms de Romilly 158 stated that a p -\-b p = c p , p a prime >2, implies 



a = x+y, 



Mp v+1 q u+1 = 2"W - 1 , 2""a a = P + Q, 



p and q odd and relatively prime, q> 1, and u, v, 9, ju, a integers ^0. [Since 

 c b = y is a power of p, Fermat's equation is impossible by Abel's 16 result.] 



153 Jahresbericht d. Deutschen Math.-Vereinigung, 4, 1894-5, 517-25. French transl., 



Annales Fac. Sc. Toulouse, [(3), 1, 1909;] (3), 2, 1910, 448; (3), 3, 1911, for errata, table 

 of contents, and notes by Th. Got on the literature concerning Fermat's last theorem. 



154 Messenger Math., 24, 1894-5, 97-99. Reprinted, Oeuvres de Fermat, IV, 159-61. 



155 Comptes Rendus Paris, 120, 1895, 1139-43 (minor error, 1236). 



156 Ueber unbestimmte Gleichungen, 1895. 



167 L'interme'diaire des math., 2, 1895, 23; repeated, 8, 1901, 305-6. 

 IUd., 2, 1895, 281-2; repeated, 11, 1904, 185-6. 



