CHAP. XXI] SUM OR DIFFERENCE OF TWO CUBES A SQUARE. 581 



" Alauda " 237 noted that nx 2 = ?/+z 3 if x = 3n, y = 2n, z = n. E. Fauquem- 

 bergue (ibid., 6, 1899, 131) gave [Euler 225 ] 



a&{6(a 2 +36 2 )} 2 =(6ab+a 2 -36 2 ) 3 +(6a&-a 2 +3& 2 ) 3 . 



K. Schwering 238 obtained an infinity of solutions by means of the rela- 

 tion between Abel's theorem and certain diophantine equations, first 

 indicated by Jacobi 148 of Ch. XXII. Set 



x 3 + 1 (mx + ri) 2 = (x ai) (x a: 2 ) (a; as) 

 By the coefficients of x 2 and x, 



m 



aia!2 + 0:10:3 + 0:20:3 



Substitute for m, n their values from raai+n = (ai + 1) 4 for i = l, 2. Thus 



a 10:2 4 (a 1+0:2) 



Q, __ 



2 ) + 2 + 2 V( 



Hence we get w 3 +n and thus (ajj+l)*. Take 0:1 = 0-2 = 0:. Then 



a 6 +20a- 3 -8 



By eliminating 3 , w^e get the desired solution 



The corresponding Abel theorem is here 



A. S. Werebrusow 239 gave Euler's 225 final solution. 



F. de Helguero 240 solved (x-y)t = z\ where t = x 2 +xy+y 2 . Set d = 3 or 

 1 , according as t is or is not divisible by 3 . Then x y = dor, t = d(3 2 . Thus 

 d/3 2 has one of the three representations by x 2 +xy+y 2 . It remains to 

 make d(xy) = H. According as d = 3 or 1, this reduces to u 2 v 2 = w 2 or 

 u 2 -3v 2 =l/ 

 m F. Pegorier 241 discussed (z+1) 3 x 3 = D. 



A. Gerardin 242 noted that one solution of a 3 +jS 3 = y 2 implies a second since 



(a 3 +4/3 3 ) 3 -(3a^) 3 -(a 3 +/3 3 )(Q; 3 -8/3 3 ) 2 . 



W. H. L. Janssen van Raay 243 discussed the solution of x 3 +y 3 = z 2 . 

 Cashmore 244 gave the first solution due to Hoppe. 229 

 See Bouniakowsky, 135 Mordell, 176 and Baer 224 ; also Catalan 122 " and 

 Tafelmacher 160 of Ch. XXVI. _ 



237 L'intermediaire des math., 5, 1898, 75-6. 



238 Archiv Math. Phys., (3), 2, 1902, 285-8. 



239 L'intermediaire des math., 11, 1904, 153. 



240 Giornale di Mat., 47, 1909, 362-4. 



241 Bull, de math, elem., 14, 1908-9, 51-52. 



242 L'intermediaire des math., 18, 1911, 201-2. Cf. Weill. 2 ' 3 



243 Wiskundige Opgaven, 12, 1915, 67-71 (Dutch). 



244 L'intermediaire des math., 23, 1916, 224. 



