578 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



J. G. van der Corput 221 applied quadratic forms to prove the impossi- 

 bility of x 5 y 3 = p m z 3 if p is a prime =2 or 5 (mod 9). 



B. Delaunay 222 stated that, if p is an integer not a cube, pz 3 +7/ 3 = l has 

 no integral solutions if the fundamental unit u of the domain defined by 

 r= Mp is not of the form Br-\-C, but has the single solution x = B, y = C, if 

 it be of that form. Here u is Dirichlet's ar 2 +6r+c, where a, b, c are integers 

 not of like sign, whose powers, with positive and negative exponents, give 

 all the units ar z +(3r-\-y, where a, ft, y are integers. 



M. Weill 223 used the identity 



to show that, if one solution of x 3 -{-y 3 = Az 3 is known, a second is 



where a=x z , j8 = y 3 , and to obtain solutions when A =3c 2 +3c+l. 



W. S. Baer 224 proved that n can be represented in the form n = <f>(u) + <(V), 

 where 4>(x] =a 3 +7#, with u, v, a, y integers and u>%, v>, if and only if 

 n is a product of two integers : n = kl, where k>2,l = al'+y, I' < k 2 3&+3 2 , 

 I' being integral and 41' k 2 the triple of a square. Then u and v will be 

 relatively prime if and only if the g. c. d. of k and I' is 1 or 3, and in the 

 latter case V is not divisible by 3 2 . The theorem can be extended to cubics 

 3? = AX 3 -\-BX 2 -\-CX+D, where A, , D are integers and B is divisible 

 by 3A, since X=x Bf(3A) transforms 6 ($ 5) into <. In particular, 

 let a = l, 7 = 0, = 0. Then n is representable as a sum of two positive 

 cubes if and only if n is a product of two positive integers k and I such that 

 l<k 2 and 41 k 2 is the triple of a square; the cubes will be relatively prime 

 if and only if the g. c. d. of & and Us 1 or 3, and in the latter case I is not 

 not divisible by 3 2 . 



If h is a positive integer, and p is a prime or unity, u 3 -\-v* = hp v has only 

 a limited number of relatively prime positive solutions, and the remaining 

 solutions are readily deduced. But u 3 +v 3 = w 2 has an infinitude of positive 

 solutions of which u and v are relatively prime. 



L. Varchon 224 " proved that x 3 y 3 = 2 a 5 b is impossible in integers +0; 

 Moret-Blanc's 192 result is a corollary. 



M. Rignaux 2246 derived (1), (2) and analogous identities from a common 

 source. 



SUM OR DIFFERENCE OF TWO CUBES A SQUARE. 



L. Euler 225 noted that x 3 -\-y 3 -H for x = pz/r, y = qz/r, z = r z /(p 3 -}-q 3 ). 

 To obtain integers, set r = n(p 3 +g 3 ); then 



x = n 2 p(p 3 +q 3 ), y = 



221 Nieuw Archief voor Wiskunde, (2), 11, 1915, 64-8. 



222 Comptes Rendus Paris, 162, 1916, 150-1. 



223 Nouv. Ann. Math., (4), 17, 1917, 54-9. 

 22 < Tohoku Math. Jour., 12, 1917, 181-9. 

 2240 Nouv. Ann. Math., (4), 18, 1918, 356-8. 

 2246 L'intermddiaire des math., 25, 1918, 140-2. 



m Novi Comm. Acad. Petrop., 6, ad annoa 1756-7, 1761, 181; Comm. Arith. Coll., 1, 1849, 

 207; Opera Omnir, (1), II, 454. 



