CHAP. VI] SUM OF TWO SQUAKES. 239 



V. A. Lebesgue 68 proved that y 2 + 1 =1= x m if y 4= 0, m > 1, by use of 

 complex numbers. 



G. Bellavitis 69 stated that every solution of x 2 + y 2 = 5 13 17 is given 

 by 



x + yi = (2 i)(3 2i)(4 db i). 



If each d is a prime 4& + 1, x* + y* = c^c 2 has k = %(m 1 +l)(m 2 +l) - 

 or k \ essentially different sets of solutions, according as y = gives 

 no solution or a solution. 



E. Prouhet 70 proved Gauss' 37 formula. 



D. Chelini 71 gave an " elegant proof " of Gauss' formula by noting that 

 every solution of x 2 + y 2 = (a 2 + 6 2 ) m (aJ + 6j) m ' is given by the de- 

 velopment of 



x + yi = (a + U} n (a - U) m ~ n (ai + M) ni (ai - bii} mi ~ ni -, 

 where n = 0, 1, , m; n\ = 0, 1, , mi; etc. 



A. Genocchi 72 noted that Chelini 71 did not prove that the solutions 

 obtained are all different, nor that no other solutions exist. 



V. Bouniakowsky 73 proved that every prime 8k + 5 is a El by use of 

 his formula (10), Ch. X, Vol. I, involving sums of divisors. 



H. Suhle 73 " noted that Jacobi's 48 theorem implies the generalization that 

 the number of positive solutions x, y of x z + y 2 = p is the excess of the 

 number of divisors 4m + 1 of p over the number of divisors 4m -f 3. He 

 proved Eisenstein's 56 result. 



C. Hermite 74 noted that, to express as a 12] a number A for which 

 a 2 = 1 (mod A) is solvable, it suffices to consider the form 



Ax 2 + 2axy + A~ l (a 2 + l}y\ 

 which is reducible to X 2 + Y 2 . 



A. Genocchi 75 considered the number of representations of n by u 2 + v 2 . 

 By the remark of Euler 24 (end), it suffices to take n odd. Let t be the g.c.d. 

 of u, v. If n has a prime factor p = 4m + 3, set n = p*ri, t = p"t', 

 where n' and t' are prime to p. Since p cannot divide a G3, w = 2p, so 

 that the product of all the prime divisors 4m + 3 of n is a square which 

 divides u 2 and v 2 . It thus suffices to treat the case in which every prime 

 factor of n is of the form 4m + 1. For such an n, set n = p*n', p being a 

 prune not dividing n'. Then 



(u 4- iv)(u ~ iv) = (q + irY(q ir) r n', q 2 + r 2 = p. 



Now q ir are complex primes, and decomposition into such primes is 

 unique. Thus 



u + iv = i l (q + ir) h (q ir) k (u' + iV), 



68 Nouv. Ann. Math., 9, 1850, 178-181. 



69 Annali di Sc. Mat. e Fis., 1, 1850, 422-5. 



70 Comptes Rendus Paris, 33, 1851, 225-6. 



71 Annali di Sc. Mat. e Fis., 3, 1852, 126-9. 



72 Nouv. Ann. Math., 12, 1853, 235-6. 



73 M6m. Ac. Sc. St. Petersbourg, (6), 5, 1853, 303. 



730 De quorundam theoriae numerorum theorematum applicatione, Berlin, 1853, 18, 26. 



74 Jour, fur Math., 47, 1854, 345; Oeuvres, I, 237. 

 76 Nouv. Ann. Math., 13, 1854, 158-170. 



