CHAP, xxi] NUMBERS THE SUM OF Two RATIONAL CUBES. 573 



Take 



Then a 2 +36 2 = (p 2 +3g 2 ) 3 , a = 4nr 3 . Hence take p = af 3 , 

 p 3q = 2yh 3 , a^ n, fgh = r. Substituting the resulting values of p, q 

 into p = af 3 , we get af 3 = j3g 3 -\-yh 3 . If the latter be solvable, the proposed 

 equation is solvable. He noted (pp. 244-5) that 16 2 -3-23 2 = (1-3-2 2 ) 3 , 

 whereas 16+23 V3 4= (1+2 V3) 3 . In general, x~-ny z = (p 2 -nq-)* implies 



but not the relation with the first factor omitted. 



A. M. Legendre 184 proved that, for A = 2, every set of integral solutions 

 has x = y, while for A = 2 m , m>l, x= y, and observed that, for A = d=3 

 or 4 (mod 9), z must be divisible by 3. He stated that the equation is 

 impossible for A = 3, 5, 6, whereas for A = 6 it has the solutions 185 x = 37, 

 y = l7,z = 2l. 



On geometrical aspects of the problem, see Glenie, 12 Becker. 16 

 Wm. Lenhart 186 gave a table of 11 pages expressing 2581 integers 

 < 100000 as a sum of the cubes of two positive rational numbers. Formulas 

 used in the construction of the table were deduced as follows from 



x*+y*=(x+y)Q, Q = x 2 -xy+y 2 . 



First, let x-\-y = a 3 , x>y, where a is even. For .7 = 1, 2, 3, , take x 

 y = sj, 2s = a 3 . Then 



the successive values of 3j 2 being computed by their differences. For a 

 odd, take x = s -\-j, y = s(j l); the new right member is s 2 + s + 3j 2 3j + 1 . 

 Similarly for x+y = a'a? or 9a'a 3 . Next, let Q = m 3 . Then 



whence 



with three similar formulas. Euler's 6 solution (Ch. XX) of Q = m* is 

 quoted. Finally, let Q m'm?} then 



F = a 2 m 3 +aa'(x+y)+a' 2 m', 

 from which is derived four similar formulas whose right members have 



184 Th<orie des nombres, Paris, 1798, 409; Mem. Acad. R. Sc. de 1'Inatitut de France, 6, 

 anne 1823, 1827, 51, p. 47 (=pp. 29-31 of Suppl. 2 to ed. 2, 1808, of ThSorie des 

 nombres). This Supplement is reproduced in Sphinx-Oedipe, 4, 1909, 97-128; errata, 

 5, 1910, 112. Theorie des nombres, ed. 3, 2, 1830, 9. 



185 G. Lame", Comptes Rendus Paris, 61, 1865, 924. 



186 Math. Miscellany, Flushing, N. Y., 1, 1836, 114-128, Suppl. 1-16 (tables). 



