574 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xxi 



the factor F. In the continuation (pp. 330-6), it is noted that 



(s'x+r'm*\ 3 , (s'y-r'm*\ 3 , 2 , 2 



I =s (z+w){3r m 3 +3sV(;c ?/)+s' m'} 



\ m J \ m / 



if Q = m'm?. If also x +y = a 3 , we may simplify this formula. To apply to 

 (A), divide each member by a 3 and set (s 2 -j-3j 2 )/w 3 = m'; hence 



( ^'+j)+ f> '' ) 3 + ( !^L-J^_ 3 Y . s w V+OrY.j+.'W). 

 \ am / \ am / 



G. L. Dirichlet 187 proved by descent the impossibility of r ? ?/ 3 = 42 3 - 

 Hence z 3 ?/ 3 = 2"z 3 is impossible, having been proved by Euler for n = 0, 

 n=l. 



J. P. Kulik 188 tabulated the odd numbers to 12097 (to 18907) which 

 are differences (sums) of two cubes, and gave the cubes. 



J. J. Sylvester 189 stated that there are no solutions for A =2, 3. He 190 

 proposed the question: If p and q are primes of the respective forms 

 18Z+5 and 18Z+11, it is impossible to decompose p, q 2 , 2p, 4q, 4p 2 , 2q 2 

 into a sum of two rational cubes. 



C. A. Laisant 191 proved that a 3 -6 3 = 10 n H f-10 n * is impossible if 



k = 3, 4 or 5. 



Moret-Blanc 192 stated that a?-b 3 = h-W n is impossible if h = l, 2 or 8. 



T. Pepin 193 proved that, if p and q are primes of the respective forms 

 18Z+5 and 18Z+11, the equation is impossible when A = p, p z , q, g 2 , 2p, 2q 2 , 

 4p 2 , 4g, 9p, 9g, 9p 2 , 9g 2 , 5p 2 , 5g, 25p, 25<? 2 . If the sum or difference of two 

 numbers is a cube, their product is expressible algebraically as the sum of 

 two cubes. Hence the double of a triangular number is a sum of two 

 rational cubes. Since a prime 6m +1 is of the form A 2 +3S 2 , it is a sum 

 of two rational cubes if one of the three numbers 2 A, 3BA is a cube, or 

 if 2B or A 5 is the triple of a cube. 



Pepin 194 proved that Euler's and Legendre's use of numbers a+6V^3 

 is legitimate and hence showed that the equation is impossible for A = 14, 

 21, 38, 39, 57, 76, 196, and stated that it is impossible for 31, 93, 95, 190. 



E. Lucas 195 noted that a solution x, y, z yields the solution 



Z = x 9 -? 



For A = 9, we get 919, 271, 438, and in general all solutions with z 

 even (not given by Prestet, Euler, Legendre). For A =7, we get 196 73, 



'" Werke, II, Anhang, 352-3. 



188 Tafeln der Quadrat- und Kubik-Zahlen aller ZahJen bis Hundert Tausend . . . , Leipzig, 



1848. 



189 Annali di Sc. Mat. e Fis., 7, 1856, 398; Math. Papers, II, 63. 



190 Nouv. Ann. Math., (2), 6, 1867, p. 96. 



191 Ibid., (2), 8, 1869, 315. J. Joffroy stated that a 3 -6 3 = A;-10 n is impossible. 



192 Ibid., (2), 9, 1870,480. 



193 Jour, de Math., (2), 15, 1870, 217-236; Extract, Sphinx-Oedipe, 4, 1909, 27-8. Proof for 



P, P 2 , Q, <f by Hurwitz, 312 p. 220. 

 1M Jour, de Math., (3), 1, 1875, 363-372. 



196 Bull. Bibl. Storia Sc. Mat., 10, 1877, 174-6. Nouv. Corresp. Math., 2, 1876, 222. 

 198 Stated by Lucas, Nouv. Ann. Math., (2), 15, 1876, 83. 



