536 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xx 



E. Fauquembergue 22 gave an insufficient proof that z 3 +24=2/ 2 if #=}= 1. 

 C. Stormer 23 solved x 2 y z = z* by means of the identity 



{x(x*+3y*} } 2 - {y(y*+3x^ } 2 = (z 2 -?/ 2 ) 3 . 



A. Goulard 24 proved that x 2 1=2 3 only for 2 = 9, since x 2 l = 8wr 3 has 

 no solution except when w = or 1 [Legendre, 81 of Ch. I]. T. Pepin (pp. 

 283-5) reduced the question to u s -\-x* = 2y z which holds only for u = x 

 [Legendre, Theorie des nombres, ed. 2, 1808, 347]. 



E. de Jonquieres 25 treated x 2 -a? = y\ For a = 3, E. B. Escott 26 noted 

 the solutions y= 2, 0, 3, 6, 40 and stated that there are no others < 1155. 



Concerning Fermat's assertion that 25 is the only square which in- 

 creased by 2 gives a cube, H. Delannoy 27 remarked that Euler's 6 proof is 

 incomplete since if applied to x 2 +47 = z z it yields x 500 but not the solution 

 # = 13. P. Tannery 28 replied that the proof as given by Legendre 7 depends 

 on the fact that every divisor of x 2 +2 is of the form p 2 +2g 2 , while not 

 every divisor of # 2 +47 is of the form p 2 +47# 2 . I. Ivanoff (p. 47) explained 

 the difference by the fact^that in the domain R(^^2) of the complex 

 integers depending on V^2 the introduction of ideals is superfluous, but 

 not for E(V-47). E. Landau 29 supplemented Ivanoff 's remark by noting 

 that a second circumstance is necessary to justify Euler's conclusion that 



(x+ AP2)(~ V :r 2) = ^ 3 implies that x V^2 are cubes in E(V^2), viz., 

 that, in R( ^2), 1 are the only units (complex integers dividing unity). 

 From the superfluity of the introduction of ideals, we can conclude only 

 that, if a product of two relatively prime complex integers is a cube, each 

 of the two factors is a product of a cube by a unit. For R( V2), the intro- 

 duction of ideals is unnecessary, but (x+ ^2)(x \/2) =P does not imply 

 that z V2 are cubes of integers a+/3 V2. Cf. Euler 183 of Ch. XXI. 



A. Boutin 30 stated that x*-7y* = l for x = l, 2, 4, 22, but for no other 

 values <196. Other writers 31 stated that any new solution has at least 

 1400 digits in y. 



E. Fauquembergue 32 noted that px 2 -\-mxy-\-qy 2 = z* for 



- mpqg 3 ) , y = p~g{ 3/ 2 + 3mfg + (ra 2 - pq) g 1 } , 



T. Pepin 33 remarked that all the theorems in his 19 " 21 papers on insolvable 

 equations # 2 +q/ 2 = 2 3 were subject to the restriction that z is odd. The 

 enunciation of this restriction is necessary if c = Sl or 81+7 since in these 



22 Mathesis, (2), 6, 1896, 191. Criticized by L. Aubry, 1'intermcd. des math., 18, 1911, 204. 



23 L'intermcdiaire des math., 2, 1895, 309. 



24 Ibid., 3, 1896, 135. 



28 Ibid., 6, 1899, 91-5; 5, 1898, 257 (a = 3). Cf. Descartes, 14 Ch. XIII; Tait, 26 Ch. XXI. 



26 Ibid., 7, 1900, 135. 



27 Ibid., 5, 1898, 221-2. 



28 Ibid., 6, 1899, 48. 



29 Ibid., 8, 1901, 145-7. 



30 Ibid., 8, 1901,278. 



81 Ibid., 11, 1904, 44 (9, 1902, 109, 183-5). 

 Z2 Ibid., 9, 1902,311-2. 



33 Ann. Soc. Sc. Bruxelles, 27, II, 1902-3, 121-170. Extract in Sphinx-Oedipe, 5, 1910, 

 10-13 (of nume'ro special), 42-6. 



