CHAP, xxi] EQUATIONS OF DEGREE THREE IN Two UNKNOWNS. 599 



E. Maillet 350a discussed ?/ 3 y = c*(x 3 x), where c is rational. For each 

 value of c there is only a finite number of integral solutions. 



Solutions 3506 have been found for the equation in binomial coefficients 



fu+l\ fv+l\ fw+l\ 



( 3 ) + ( 3 ) = ( 3 )> u*-u+v*-v=w*-w. 



The sum of the first n odd cubes can 350c be expressed as a sum of seven 

 squares 4=0. Special solutions of x*+y 3 +z 3 = k(x+y-\-z) are noted (p. 155). 



On I(mp 2 +nq 2 ) = X(mr 2 +ns 2 ), where I, X are linear functions of p, q, r, s, 

 see papers 48, 51, 55, 80, 89. On tu 2 +t 2 v = Auv 2 , see Lagrange, p. 572. On 

 <(-w)-i-0(y)=<7, see Baer. 224 



On x(l-x 2 ) = Ay 2 , see Tweedie 74 of Ch. IV. 



On x+a/x = y 2 , see Leibniz 64 and Terquem 70 of Ch. XXII. On equations 

 of degree three involving products of consecutive numbers, see papers 28, 32, 

 56, 58, 59, and 63 of Ch. XXIII. On xy(x+y) = Az\ see Euler 10 , Lucas 199 , 

 Catalan, 204 and Hayashi 219 ; also Lucas 150 of Ch. I. Chuquet 34 of Ch. XII 

 expressed 20 as a sum of three positive rational cubes; on the general topic, 

 see papers 404-29; also Ch. XXV, end. On x 2 +y 2 -}-z~ = kxyz, see the 

 papers cited under Hurwitz 174 of Ch. XXIII. 



SYSTEMS OF EQUATIONS OF DEGREE THREE IN TWO UNKNOWNS. 



Diophantus, IV, 29, 30, made xy (x+y) cubes. Take y = x-x. Then 

 the condition with the upper sign is satisfied and that for the lower sign 

 requires x 3 2z 2 = cube = (%x) 3 , say, whence x = 16/7. 



Bombelli 351 treated the same problem. 



Bhascara 352 noted that the sum and difference of 4y 2 and 5y 2 are squares 

 and their product 20?/ 4 is a cube, (10?/) 3 , if y = 50. The sum of the cubes of 

 y- and 2y 2 is 9?/ 6 , a square, and the sum 5?/ 4 of their squares is a cube, (5?/) 3 , 

 if y 25. Under Bhascara 30 of Ch. XII is given his solution of xy= D, 

 x 2 -\-y z =z 3 , and of y 2 +z 3 = D, y-\-z= D. 



L. Euler 353 discussed x+y=H, x 2 +y 2 = p 3 . Hence take p = a 2 +6 2 , 

 z = a(a 2 -36 2 ), y = b(3a 2 -V). Then x+y = (a-b)Q, Q = a 2 +4ab+b 2 . Set 

 a-b = c 2 . Then Q = 66 2 +66c 2 +c 4 = (c 2 +36//^) 2 if b/c 2 = 2g(g-f}/(3f 2 -2g 2 ). 

 Then x and y will be positive if b = 2g(gf), c 2 = 3/ 2 2g 2 . The latter is 

 satisfied if /=11, g = l, c = 19, or if /= 3, g = l, c = 5, whence 6 = 8, a = 33, 

 a; = 29601, y 25624. For three numbers he gave only results: 

 35+9+5 = 7 2 , 35 2 +9 2 +5 2 = ll 3 ; 67+9+5 = 9 2 , 67 2 +9 2 +5 2 = 19 3 . 



[But the last sum equals 5-919 4= 19 3 .] 



W. Spicer, 354 to find two squares whose sum is a square and difference a 

 cube, took a = ^x 2 +%x 3 and b = %x 2 |x 3 as the squares with the sum x 2 



. Ann. Math., (4), 18, 1918, 289-292. 

 3606 Zeitschrift Math. Naturw. Unterricht, 50, 1919, 95-6. 

 350c L'intermediaire des math., 26, 1919, 77-8, 109-10. 



361 L'algebra opera di Rafael Bombelli, Bologna, 1579, 553. 



362 Vija-ganita, 121-2. Colebrooke, 3230 201-2. 



363 Opera postuma, 1, 1862, 255-6 (about 1782). 



3M Ladies' Diary, 1766, 33-4, Quest. 536; C. Button's Diarian Miscellany, 3, 1775, 220; Ley- 

 bourn's M. Quest. L. D., 2, 1817, 251. 



