CHAP. XXI] DlOPHANTINE EQUATION OF DEGREE THREE. 597 



either m = 5t, whence m = 5, n = 9, or m = 20, n 58, if m < 300 ; or m = 25k -f- 1 , 

 whence m = l, n = l, if m<1000. 



Realis 3330 noted that the double of any square, as well as the triple of 

 the square of any even number >2, equals the excess of a sum of two 

 squares over a sum of two cubes. 



M. Weill 334 noted that (1) has the solution x = pA, y = hp 3 !, z = px, 

 u=hpy, where A = ph-+l; also, x = HA 2 , y=AB, z = H 2 A, u = hHB, 

 where H = h 3 p, B = p 3 h-\-3ph~ h 5 +l. The last solution is based on the 

 identity A 3 -hH 3 =(l+h 5 )B. 



H. S. Vandiver and W. F. King 335 proved the impossibility of 



G. Bisconcini 336 noted that x 3 y 3 = (x+y) 2 has the single solution x = l, 

 y = Q, in integers; x 3 +y 3 = x 2 +y 2 has only the solutions x = l, y = or 1; 

 (xyY xy or x 2 +y 2 has various solutions. 



References 337 on cubic equations with integral roots are in place. 



A. Cunningham 338 noted that one method of solving x 3 +y 3 = z 2 +u 2 is 

 to make x+y and x 2 xy+y 2 both sums of two squares. 



A. Gerardin 339 satisfied Sz 3 = S?/ 2 by taking 



and equating the coefficients of m 2 (thus determining g), so that m is found 

 rationally. Another method is to take g 0. 



R. Norrie 84 noted that from one set a i} - , a n of solutions not all zero 

 of a homogeneous cubic equation in Xi, , X n we can in general deduce 

 further sets by substituting Xi = rxi+di (i = l, , n), thus deriving 

 ar 3 +j8r 2 +T/* = 0. Since 7 is linear, we can make 7 = by choice say of x n 

 in terms of xi, - -, x n -i. Then take r= (3 /a. The method is applied to 

 bx(x 2 -V) =u 2 +2v 2 and to 



As to this method see Lagrange 324 and the related method of Cauchy 287 

 and Lucas. 296 



A. Cunningham and E. B. Escott 340 made xy(x+y)+l a cube, where 

 l = xy or 2x-\-2y; also xy2(x-+-y) is made a cube. 



Welsch 341 noted that 1, 2, 3 are the only three positive integers whose 

 sum equals their product. For n integers see papers 150-2 of Ch. XXIII. 



A solution 342 of 2x< ^y\ = 2*4 is x i} yi = ^(u*Ui). This and other 

 solutions are found by decompositions of u 3 = x 2 y 2 . 



33 3o Nouv> Ann> Math., (3), 2, 1883, 295-6. 

 334 Nouv. Ann. Math., (3), 4, 1885, 184-8. 



335 Amer. Math. Monthly, 9, 1902, 293-4; 10, 1903, 22. Cf. Euler 9 ; also Hurwitz 212 of Ch. 

 XXVI. 



336 Periodico di Mat., 22, 1907, 125-9. 



337 L'interm6diaire des math., 15, 1908, 47-8, 152, 239; 16, 1909, 208. 



338 Ibid., 18,1911,210-3. 



339 Bull. Soc. Philomathique, (10), 3, 1911, 226-233. Cf. paper 285 above. 



340 L'interme'diaire des math., 19, 1912, 164-5, 273. 



341 Ibid., 69. 



342 Ibid., 20, 1913, 190, 239-40. 



