CHAP, xxvi] FERMAT'S LAST THEOREM. 751 



term. Hence n divides x or y. For, if x+y and hence z is divisible by n, 

 set x = z+nk y in the initial equation; the result is said to hold only if y 

 is a multiple of n. 



F. Fabre 111 proposed the question of the divisibility of (x+y) n x n y n 

 by x 2 +xy+y* and M. Dupuy proved (ibid., 1881, 282-3) that n must be 

 of the form 6al. 



If 112 (Za) 2n+1 = 2a 2n+1 is true for n= I it is true for any n, since 



A. E. Pellet stated and Moret-Blanc 113 proved that 

 (mod 7) is solvable if ABC is prime to 7. 



E. Cesaro 114 proved that if \l/(ri) is the number of sets of positive integral 

 solutions of Ax a +By ft =n, where A and B are positive integers, 



The ratio of \f/(n) to the number of solutions of x a +y fi = n is A~ 1/0 5~ 1//J , 

 in mean. Hence, for a = (3 = l, \j/(n)=n/(AB), in mean. For a = /3 = 2, 

 lKw)=ir/(4VZB), in mean. The mean of the sum of the pth powers of 

 all the positive integral values which x can take in x k -\-y k = n is found 

 (p. 229). 



C. M. Piuma 115 noted that, if no one of the coefficients A, B, C is divisible 

 by the prime m = pq+l, then Ax p +By-\-C=Q (mod m) has integral 

 solutions if and only if Az -\-Bzi +(7=0 (mod ra) has solutions for which 

 z^x p , Zi=y q are solvable for x, y, i. e., if 



s(z-l)=0, zi(2?-l)=0 (mod m) 



are solvable. Thus the initial congruence has solutions if and only if 

 P=0 (mod m), where P is the resultant of the equations corresponding to 

 the last two and Az+Bzi+C = Q, so that P is a product of (p+ !)(<?+!) 

 linear factors. 



For q = 2, there are solutions if C+A or C A is divisible by m, or if 

 any one of the products BC, B(C-\-A), B(CA) is a quadratic 

 residue of m. In particular, Ax z -\-By*+C=Q (mod 7) is solvable if no 

 one of the coefficients is divisible by 7. Cf. Pellet. 113 



E. Catalan, P. Mansion and de Tilly 116 gave adverse reports on two 

 manuscripts submitted for the prize offered for 1883 by the Belgian Academy 

 (p. 101) for a proof of Fermat's last theorem. 



E. de Jonquieres 117 proved that in a n -\-b n = c n , n>l, the greater of a, b 

 is composite. Set c = a+k, b>a. Then, by the binomial theorem, 



111 Jour, de math. elementaire de Longchamps et de Bourget, 1880, No. 273, p. 528. 



112 Math. Quest. Educ. Times, 36, 1881, 105. 



113 Nouv. Ann. Math., (3), 1, 1882, 335, 475-3. 



114 Mem. Soc. R. Sc. de Liege, (2), 10, 1883, No. 6, 195-7, 224. 

 118 Annali di Mat., (2), 11, 1882-3, 237-245. 



" 8 Bull. Acad. R. Belgique, (3), 6, annexe 52, 1883, 814-9, 820-3, 823-32. 



117 Atti Accad. Pont. Nuovi Lincei, 37, 1883-4, 146-9. Reprinted in Sphinx-Oedipe, 5, 1910, 



29-32. Proof by S. Roberts, Math. Quest. Educ. Times, 47, 1887, 56-58; H. W. Curjel, 



71, 1899, 100. 



