240 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vi 



where the final factor divides n'. Multiplying by the conjugate, we get 

 n = p h+k (u' 2 + v' 2 ). Hence h + k = TT, n' = u' 2 + v' 2 . The multiplica- 

 tion of u + iv by i~* at most interchanges u 2 and v 2 . Hence the effective 

 solutions u, v are given by 



u + iv = (q + ir) h (q - - irY~ h (u' + iv'} (h = 0, 1, , TT), 



where u', v' range over the N' solutions of u' 2 + v' 2 = n'. If we change 

 the sign of v' and replace h by TT h, we get u iv. If TT is even, and n' 

 is a square u' 2 , the representation n = (p ff/ V) 2 is excluded. Hence the 

 number of representations of n as a E] is ^(TT + 1)N', unless TT is even and n 

 is a square, and then is f { (TT + 1)N' 1 } . The number of representations 

 of n' is %N' or %(N f 1) according as N f is even or odd. Hence by induc- 

 tion we obtain Gauss' 37 result that if a, b, - are distinct primes 4m + 1, 

 the number of representations of n = a a b^ as a E] is %N or %(N 1), 

 according as n is or is not a square, where AT = (a + l)(/3 + 1) *. The 

 second would be %(N + 1) if we count also the case of n + 0. Hence the 

 " correction " by Volpicelli 67 is unnecessary. 



P. Volpicelli 76 retracted his 67 claim of an error on the part of Gauss 37 

 and Legendre, 38 but gave k % as the number of representations of M 

 as a E] when ju and a, 0, are all even, i. e., when M itself is a square. 

 Concerning Euler's remark, quoted by Genocchi, 75 that an integer and its 

 double have the same number of representations as a E], Volpicelli (p. 185) 

 stated that p = 4225 has only four [omitting p = 65 2 + 0], while 2p has 

 five, representations. 



A. Genocchi 77 answered the latter objection by noting that zero is to be 

 counted as an integer. He remarked (p. 495) that the " new " case noted 

 by Volpicelli (that of M a square) had been treated by Fermat, who dis- 

 cussed the number of ways a number is the hypotenuse of a rational right 

 triangle. 



A. Cayley 78 noted that a formula of Dirichlet's 52 becomes, for D = 1, 



(1 + 2(? 4 -f- 2g 16 + 2g 36 + -)(g H~ g 9 ~t~ g 25 H~ ) 



q g 3 g 5 g 7 i 



= 1 - q 2 " 1 - g 6 "r^^" 1 - g 14 ' 



H. J. S. Smith, 79 in accord with Gauss 24 of Ch. II, denoted by [gi q n ~] 

 the numerator of the common fraction equal to the continued fraction 



1 1 1 



gi "i" , ' i - > 



qz + #3 + + q n 



and employed Euler's 72 relations (Ch. XII) 



(3) [g! g n ] = [ gl . . . g J[g t - +1 g n ] + [gl gi-l][g+2 ' ' ' gn]. 



78 Annali di Sc. Mat. e Fis., 5, 1854, 176-186; Jour, fur Math., 49, 1855, 119-122. 



77 Annali di Sc. Mat. e Fis., 5, 1854, 491-8. 



78 Cambridge and Dublin Math. Jour., 9, 1854, 163-5. 



" Jour, fur Math., 50, 1855, 91-2; Coll. Papers, I, 33-4. Reproduced by Borel and Drach, 

 Introduction a la thdorie des nombres, 1895, 109-12; Chrystal, Algebra, ed. 1, II, 1889, 

 471; ed. 2, II, 499. 



