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



Euler 51 deduced Vieta's formula (2) and noted that in (6) the second 

 factors have a common divisor of like form 2 +3w 2 . From 



(7) p 2 +3g 2 = (/ 2 +3 2 )(* 2 +3O, s 2 +3r 2 = (/i 2 +3& 2 )(i 2 +3O, 

 he concluded that 



(8) p=ft+3gu, q = gtfu, s = ht+3ku, r = kthu. 



Inserting the values of p, s and (7) into (6) and deleting the common 

 factor / 2 +3w 2 , we obtain tfu rationally. To avoid fractions, take u equal 

 to the denominator. Thus 



(9) u=f(f*+3g*)-h(h*+3k*), t = 3k(V+3k^-3g(f 2 +3g^. 



For/, g, h, k arbitrary, formulae (5), (8), (9) give the general solution of (4). 

 Special cases are 



7 3 +14 3 +17 3 = 20 3 , ll 3 +15 3 +27 3 = 29 3 , ! 3 +6 3 +8 3 = 9 3 , 3 3 +4 3 +5 3 = 6 3 . 



W. Emerson 52 repeated Vieta's discussion and treated the problem to 

 find three cubes whose sum is both a cube and a square. Cf. Hill 84 of Ch. 

 XXIII. 



J. P. Griison 53 gave (1). 



S. Jones 54 deduced (1) and (2). 



J. R. Young 55 passed from (4) to (6) as had Euler. Set p = m 2 , s = ri 2 . 

 Then (6) becomes 



3n 6 +9rV - 3m 6 = 9m 2 g 2 = (c - 3m) 2 , if r = c2 ~ 3 f+ 3m \ 



Qcn 



Take m = l, n = 2, c = 3d and drop the common denominator 4oL Hence 



He also solved (4) by taking 56 A = m l, B = n 2 p, C = ri'-{-p, D = m+l, 

 whence 9m 2 = Cn 2 p 2 +3(n 6 1) = (q 3np) 2 , gay. Hence 



np, m= 

 Multiplying the resulting values of A, - -, D by Qnq, we get 



A, D = n{g 2 =F6g+3(n 6 -l)J; B, C= Tg 2 +6?i 3 g3(n 6 -l). 



F. T. Poselger 57 treated the transformation of a sum or difference of 

 two cubes into a difference or sum of two positive cubes. 

 J. P. M. Binet 58 expressed Euler's 48 solution of 



(10) 



61 Algebra, 2, 1770, arts. 245, 248; French transl., 2, 1774, pp. 351, 360. Opera Omnia, (1), 



I, 490-7. 

 " A Treatise of Algebra, London, 1764, 1808, 382-4. 



63 Enthiillte Zaubereyen und Geheimnisse der Arith., Berlin, 1796, 125-8, and Zusatz at end 



of Theil I. 



64 The Gentleman's Diary, or Math. Repository, London, No. 90, 1830, 38-9. 



66 Algebra, 1816, S. Ward's edition, 1832, 351-2. Reproduced, Math. Mag., 2, 1895, 154-5. 



68 Reproduced, Math. Mag., 2, 1898, 254. 



" Akad. Wiss. Berlin Math. Abhandl., 1832, 27-31. 



"Comptes Rendus Paris, 12, 1841, 248-50. Reprinted, Sphinx-Oedipe, 4, 1909, 29-30. 



