562 HISTORY OF THE THEORY OP NUMBERS. [CHAP, xxi 



A. S. Werebrusow" gave the solution 72 



in which M 2 +MJV+N 2 = 3co 2 <^, W 3 = l. He 100 noted that 



(2) 

 holds for 



xly-x-yi 



, 2/2 = 



and the values derived from the latter by interchanging Xi, y\. He 101 

 used this result to get the general solution of (2). 



Fauquembergue 102 remarked that the last formula follows from the 

 identity 



due to A. Desboves 103 , by taking x?+y z =xl+yl and dividing the result by 

 the product of (xyxiyi) 3 by x*x\ = y\y z . 



A. Gerardin 104 stated that the least solution of (2) in integers >1 is 

 probably z = 560, y = 70, Zi = 552, 2/1 = 198, z 2 = 525, 7/ 2 = 315. 



Fauquembergue 105 noted that if Cauchy's 287 formulas are applied to 

 = lQz 3 , which has the solution x = 3, y= 2, 2 = 1, we get 



so that 19- 36351 3 is a sum of two positive integral cubes in various ways. 



SOLUTION OF 2(x 3 -\-z 3 )=y*-\-t 3 . 



R. Amsler 106 noted the solution x = u n +i,z = v n , y = u n +u n +i, t = v n +Vn+i, 

 where u n and v n are the nth coefficients of the developments of 



A. Gerardin 107 noted the identities 



Gerardin 108 gave several solutions, as 

 2a(a 3 -c 3 ), ?/ = c(c 3 -4a 3 ), g = b(2a 3 +c 3 ), t = 



99 L'intermSdiaire des math., 9, 1902, 164; 11, 1904, 288; Matem. Shorn. (Math. Soc. 



Moscow), 25, 1905, 417-37. 



100 L'intermMiaire des math., 12, 1905, 268; 25, 1918, 139, for numerical examples in which 



X2 and 2/2 are integers. 



101 Matem. Shorn. (Math. Soc. Moscow), 27, 1909, 146-169. 



102 L'interme'diaire des math., 14, 1907, 69. 



103 Nouv. Ann. Math., (2), 18, 1879, 407. Special case of Desboves. 302 



104 L'interme'diaire des math., 15, 1908, 182; Sphinx-Oedipe, 1906-7, 80, 128. 

 106 Sphinx-Oedipe, 1906-7, 125. 



106 Nouv. Ann. Math., (4), 7, 1907, 335. Proof by L. Chanzy, (4), 16, 1916, 282-5; same in 



Sphinx-Oedipe, 9, 1914, 93-4. 



107 Sphinx-Oedipe, 1910, 179. 



108 Ibid., 9, 1914, 143-4; Nouv. Ann. Math., (4), 16, 1916, 285-7, where Y, Z should be 



interchanged. 



