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



L. Aubry 343 noted solutions, involving two parameters, of 



xyz (z 2 +2/ 2 +z 2 ) w+4w 3 = 0. 

 Special solutions (p. 207) are given for 6 = 7, 61, 2281, 99905 of 



L. Aubry 344 treated z 3 -\-x-\-y 3 +y = z 3 +z by setting x+y = 2u, xy = 2v, 

 z = pu, whence 2(u 2 +3y 2 +l) = p(p 2 w 2 +l), which is solved as a Pell equa- 

 tion in u, v. 



E. B. Escott 345 treated the preceding problem by setting y = x+d, 

 z = x+b, x = k(bd), and found eight sets of solutions. Next, for 



,z = ex,a=x+b,d = b+ke. The discriminant of the resulting 

 equation for x must be a square, 9s 2 . Thus k = 3n 1. For n = 0, 



2x = e-b, 2y = b-e, 2z = 2a = b+e. 



For n=l, we get the solution (in which p is a rational parameter) 

 2e-R- P , Sy=-lQe-R+p, 4z = 2e+R+p, 



He gave (pp. 126-7) solutions of each of the equations x*xy+y 3 = z 2 , 

 X *x 2 y 2 +y* = z 2 . L. Aubry (p. 47) reduced x*xy+y*=z 2 to a Pell equa- 

 tion by setting %(xy) = v, u. 



A. Gerardin 346 noted that, if a?-V=f 2 -g 2 , then 



becomes a quadratic equation for m. By equating to zero one of the three 

 coefficients, we find new solutions of x 3 y* = F 2 G 2 . Cf. Cunliffe 325 ; also 

 Realis 17 - 18 of Ch. XX. 



P. Bachmann 347 solved & 3 (pl+pl-\-pl)k = 2p 1 p 2 p 3 in positive integers. 

 We may assume that pi = hiki (i = l, 2, 3), k=fkik 2 k 3 , where /=! or 2. 

 Multiplying the given equation by fkl, we get 



The factors on the left are equated to ns* and nsl respectively, by use of 

 solutions of x z h 2 = ns 2 . 



Cashmore 348 stated erroneously that x*+y* = u 2 +v 2 for 



x, y = 2(a 2 +b 2 2eh2fg}, u = 4:(a*-ab 2 +2beg+6bfh'), 



R. Goormaghtigh 349 solved a: 3 +2x+i/ 3 = D. 



T. Hayashi 350 proved that x 2 y+y 2 z+z 3 = Q is impossible in integers 



343 L'intermediaire dee math., 20, 1913, 95. 



344 Sphinx-Oedipe, 8, 1913, 46-7. Cf. Lenhart. 328 



346 Ibid., 123-4. 

 46 Ibid., 14. 



347 Archiv Math. Phys., (3), 24, 1915, 89-90. 

 848 L'intermediaire des math., 23, 1916, 224. 

 349 Ibid., 200-1. 



860 Nouv. Ann. Math., (4), 16, 1916, 161-5. 



