404 HISTORY OP THE THEORY OF NUMBERS. [CHAP, xm 



A. Sykora 25 repeated Claude's 21 first remark. 



L. P. da Motta Pegado, 26 A. Z. Candido, 26 T. H. Miller, 27 G. Bisconcini, 28 

 and H. E. Hansen 29 stated known results. 



" H. Rifoctitlee " 30 noted that every integer N is the quotient of two 

 differences of two squares. For, N = 2(a? 6 2 ) or a 2 & 2 according as 

 N=2 (mod 4) or not. Then apply formula (11) of Euler, 66 Ch. XII, for e = 1. 



W. Sierpinski 31 proved that the number r(n) of distinct representations 

 of a positive integer n as a difference of two squares is twice the difference 

 between the number of even and odd divisors of n. Also 



* r- r r _i ~i r r _i_i "i (z-D/2 



0(oO=Er(n)=2[V?]-2 +4 E [>+n a l 



n>0 L A J L Z J n>0 



where \f\ is the greatest integer ^t. If Q(ri) is the number of divisors of n, 



x z/2 z/4 I m 



0(s)=2 E 0(K)-2 E 0(2&)+2 E W, Iim- {r(fc)-0(&) } =0. 



*>0 *>0 *>0 m-a,nik=l 



S. Guzel 32 proved that 



1 



n 



EM*) -*(*)} 



Vn 



*A. L. Bartelds 320 discussed x 2 y z = g. 



For solutions of x 2 -l = g, see Stormer 274 of Ch. XII. Cf. Gill. 34 



SOLUTION OF ax 2 +bxy+cy 2 = dz z . 



Diophantus, IV, 10, desired two cubes the ratio of whose sum to the 

 sum of their sides is a square. Taking s and 2 s as the sides, we must 

 have 4-6s+3s 2 = D, say (2 -4s) 2 , whence s = 10/13. 



Diophantus, IV, 11, 12, solved x*y 3 = xy. Take x = rz, y = sz. Then 

 (r 3 s 3 )/(rs) is to be a square. For the upper signs he found (as in IV, 

 10) that r = 5, s = 8, 2 = 1/7. For the lower signs, take r = s+l, so that 

 3s 2 +3s+l = D, say (l-2s) 2 , whence s = 7, 2 = 1/13. 



In these three problems, Diophantus made no use of the fact that 

 (x 3 y 3 )/(xy) =x i ^rxy-\ry i - But, in V, 7, he made z 2 +z+l the square of 

 x-2 for jc=3/5, whence 3 2 +3-5+5 2 = D. 



C. G. Bachet in his comments solved similarly /=p 2 or 3p 2 , where 

 f=x z xy+y z . Fermat (Oeuvres, III, 249) remarked that we can solve 

 fa, where a is the product of a square by one or more primes of the 

 form 3n+l or 3. 



L. Euler 33 proved that if fx^-{-gxy-\-hy 2 = tz 2 is solvable for t = k, it is 

 solvable for t = kl, where I = p 2 +gpq+fhg'*. We have only to multiply the 



Archiv Math. Phys., 61, 1877, 446-7. 



M Jornal de Sc. Math, e Ast., 1, 1878, 150-5, 171-2. 



" Proc. Edinburgh Math. Soc., 9, 1890-1, 23-5. 



88 Periodico di Mat., 23, 1908, 21. 



29 L'enseignement math., 18, 1916, 48-55. 



80 L'interm&liaire des math., 11, 1904, 25-6. Proof, 8, 1901, 238-40, by continued fractions. 



31 Wiadomosci Matematyczne, Warsaw, 11, 1907, Suppl., 89-110. 



32 Ibid., 111-9. 



320 Wiskundig Tijdschrift, 13, 1916-7, 207-9. 

 83 Opera postuma, 1, 1862, 209-211 (about 1771). 



