CHAP. XXI] TWO EQUAL SUMS OF TWO CUBES. 559 



H. Holden 80 obtained all integral solutions a, b, c, d of 



for such values of p that any factor of a or c, not of the form l 2 +pm 2 , is 

 a factor of both. This is true if there is a single properly primitive class 

 of quadratic forms of determinant p and if, when there are improperly 

 primitive classes, the highest power of 2 which divides Z 2 -\-prn 2 has an even 

 exponent. The conditions hold for p = l, 2, 3, 5, -13, 29, 53, 

 61. For p = 3, we have the equivalent equation 



and hence the complete solution of (10). He proved that there is no integral 

 solution of the initial equation with a = b and hence none of x 3 = y 3 +z 3 . 

 J. Jandasek 81 gave the identity 



(3w 3 +3w 2 y+2w; 2 +y 3 ) 3 =(3w 2 y+2w;^ 



K. Petr 82 noted that Euler's 48 solution of x 3 +y z +z 3 = u 3 may be written 

 in the form 

 x:y:u: z 

 = A 2 E+2BC-BD:-A 2 E+BC+BD:B 2 E+2AC-AD:-B 2 E+AC+AD, 



where C, D are arbitrary and ABE~ = C~ CD-{-D 2 . It is thus not essen- 

 tially different from Binet's solution. 



Binet's 58 solution is claimed 83 to be not general. 



R. Norrie 84 treated (4) by taking A=r#i+X, J5 = ra 2 +/x, C = rx 3 /*, 

 D = rx Q +\. Thus ar 3 +3|3r 2 +37r = 0, where a = xl-x\-x\-xl, 



We may make 7 = by choice of XQ. Then ar 3 +3/3r 2 = for r= 3/3/a. 

 The resulting values of A, B, C, D in terms of xi, x 2 , x 3 , X, p are of high 

 degree and much more complicated than the complete solution by Euler 48 

 and Binet. 58 



M. Fujiwara 85 showed that the formulas by Schwering 73 and Kuhne 74 

 can be deduced by simple substitutions from formula (11) of Euler and 

 Binet. 



A. Ge*rardin 86 gave the identities 



and one similar to the latter. 



80 Messenger Math., 36, 1906-7, 189-192. 



81 Casopis, Prag, 39, 1910, 94-5. 



82 Ibid., 40, 1911, 99-102. In the Fortschritte report the sign before AD in is wrong. 



83 L'interme"diaire des math., 18, 1911, 265-6; 19, 1912, 116. 



84 University of St. Andrews 500th Anniversary, Mem. Vol., Edinburgh, 1911, 50-1. 

 86 T6hoku Math. Jour., 1, 1911, 77-8; Archiv Math. Phys., (3), 19, 1912, 369. 



86 L'interme'diaire des math., 19, 1912, 7. Cf. pp. 116-8 for references. He gave the first 

 in Assoc. frang. av. sc., 40, 1911, 12. 



