538 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. XX 



E. B. Escott 46 noted that, if if = 2x* - 1 is solvable, y = 24n 2 - 1 or 2n 2 - 1 . 



L. Aubry 47 proved that 2 + 1 +2 2fc = y 3 is impossible. If x is odd, x 2 +2 2fc 

 is a sum of two relatively prime squares, so that the factors of y 3 1 are 

 = l(mod4). Thus y 1 = 1, which gives y 2 -\-y J r 1 = 3 (mod 4). Iix2 n z, 

 where z is odd, 



Since y 2 -\-y+l is odd, its prime factors are of the form 4J+1. Thus 

 y l is divisible by 2 2 " and hence by 4. Again, y 2 -\-y+l = 3 (mod 4). 



L. Aubry and E. Fauquembergue 48 proved that 2x 2 l = y 3 has no 

 solutions other than x = 0, y= 1 ; = 1, 2/ = l; = 78, 2/ = 23. 



A. Gerardin 49 , to make G^x 2 -\-xy-\-y~ a cube, assumed that 



and took 3f+2x+y = 0. Then / and m are expressed in terms of x, y. 

 To make the result symmetrical, set y = q/3, x = p-\-q/3. Hence 



for 



X = q*+3pq*-p*, Y = -3pq(p+q), Z^ 



a result obtained otherwise by A. Desboves. 50 



Gerardin 51 treated aX 2 -{-bXY-\-cY z = hZ 3 , given one solution a, ft, 7. 

 After substituting X = a-\-mx, Y = (3-\-my, Z = y-{-mf, equate the coefficients 

 of the first powers of m (by choice of /) ; thus m is determined rationally. 



L. Aubry 52 proved that 25 is the only square which increased by 2 gives 

 a cube [Fermat 2 ~J. He 53 proved that x z +a = y* is impossible for a = 4A 2 +5 3 

 if B = l (mod 4) and A is not divisible by the square of a prime 4n 1 

 dividing B, or by 3 if B is not divisible by 3, or by 3 3 if B is divisible by 3. 

 Hence it is impossible for a = 17. 



E. Landau 54 proved that x 3 +2 = ?/ 2 has only a finite number of solutions 

 by means of Thue's result that a 3 +3o; 2 /34-6a/3 2 +2/3 3 = l has only a finite 

 number of solutions (Thue 9 of Ch. XXIII), and Landau's 29 discussion above. 



H. Brocard 55 gave eight sets of solutions of x^y z 17. 



L. J. Mordell 56 investigated y~ k = x 3 by elementary methods, by the 

 theory of ideals, and by the arithmetical theory of binary cubic forms. 

 In particular, he listed the values of k between 100 and 100 for which 

 he believed there is an infinitude of solutions. 



46 Amer. Math. Monthly, 16, 1909, 96. 



47 Sphinx-Ocdipe, G, 1911, 26-27; stated by F. Froth, Nouv. Corresp. Math., 4, 1878, 64, 223. 



48 Sphinx-Oedipe, 6, 1911, 103-4; 8, 1913, 170-1 (122-3 for E. B. Escott 's proof that a solution 



y>23 has more than 256 digits). 

 43 Assoc. franc, av. sc., 40, 1911, 10-12. 



60 Nouv. Ann. Math., (2), 18, 1879, 269, formula (8) with o = 6 = l. 



61 Bull. Soc. Philomathique, (10), 3, 1911, 222-5. 



62 Sphinx-Oedipe, 7, 1912, 84. 



63 L'intermc'diaire dcs math., 19, 1912, 231-3. 

 M Ibid., 20, 1913, 154. 



68 Ibid., 62-3. Cf . Brocard. 11 



"Proc. London Math. Soc., (2), 13, 1914, 60-80. 



