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



n_ a n j& divisible by k. But if k^b, c n ^(a+6) n >a n +6 n . 

 Hence b n is divisible by an integer k, k>l, k<b. Similarly, if a is a prime 

 <b, then c 6 = 1. He 118 stated that if a n +6 n = c n and a or 6 is a prime, 

 the least of the two is a prime and the greater is composite and differs 

 from c by unity. 



G. Heppel 119 proved that, if n is a prime >3, (x+y) n x n y n is divisible 

 by nxy(x-\-y)(x i -\-xy-\-y r } and found the coefficients of the general term of 

 the quotient. 



P. A. MacMahon 120 employed his generalization of Waring's formula in 

 Proc. Lond. Math. Soc., 15, 1883-4, p. 20, to prove the identity 



S(x, y)+S(y, *) = z(-i)w. 



summed for the integral solutions of 2a+36 = w, where 



(x-\- < 2y)x n -\-(-l} n+l (x-y)(x-\-y} n 



He gave a similar identity for three variables. The right member of the 

 initial identity becomes 5xy(x+y)(x 2 +xy-\-y 2 ) 2 if n = 7 [cf. Cauchy 29 ]. 

 E. Catalan 121 stated that if p is an odd prime, 



(x + y) p - x p - y p = pxy (x + y) P 2 , 



where P is a polynomial with integral coefficients, holds only if p = 7 and 

 P = x 2 +xy+y 2 . He 122 proved this by takings = y = l. Thus 2 p ~ l - 1 = pN 2 , 

 where N is an integer. Set t=(p 1)/2. Since 2' 1 and 2'+l are rela- 

 tively prime, having the difference 2, one of them is a square. The first 

 is of the form 4n+3 and is not a square. Hence 2*+l=M 2 . Thus the 

 factors M-f-1, Ml of 2* are powers of 2 and their difference is 2. Hence 

 M-l=2, so that p = 7, N = 3 or p = 3, N = l. 



Catalan 122 " stated the empirical theorems: (I) (x-\-l) x x x =l is im- 

 possible in integers except for a: = or 1. (II) x y y x = 1 is impossible ex- 

 cept for x=l, y = or x = 3,-y = 2. (Ill) x p 1 = P, where p and P are 

 primes, is satisfied only by x = 2, p = 3, P = 7. (IV) x n 1 = P 2 is impos- 

 sible if P is a prime. (V) x 2 l = p m , for p a prime, is satisfied only by 

 x = 3, p = 2, m=3; x = 2, p = 3, m= 1. (VI) x p q y 1, where p and q are 

 primes, is impossible except when x = y = 3, p q = 2. (VII) x*+y 3 = p 2 , 

 where p is a prime, is impossible except when re = 2, y=l, p = 3. (VIII) 

 x n = {(2 n ~ 2 l) n +l}/2 n ~ 2 is impossible except when n = 3, x=l. Cf. 

 Gegenbauer. 133 



G. B. Mathews 123 proved for special primes p that x p +y p =z p is impos- 

 sible if no one of x, y, z is a multiple of p. The method was suggested by 



118 Comptes Rendus Paris, 98, 1884, 863-4. Extract in Oeuvres de Fermat, IV, 154-5. 



119 Math. Quest. Educ. Times, 40, 1884, 124. 



120 Messenger Math., 14, 1884-5, 8-11. 



121 Nouv. Ann. Math., (3), 3, 1884, 351 (Jour, de math., sp6c., 1883, 240). 



122 Ibid., (3), 4, 1885, 520-4. 



1220 Mem. Soc. R. Sc. Liege, (2), 12, 1885, 42-3 (earlier in Catalan 90 ). 



123 Messenger Math., 15, 1885-6, 68-74. 



