CHAP. XX] BINARY QUADRATIC FORM MADE A CUBE. 535 



T. Pepin 15 applied de Jonquicres' 13 method to obtain the generalization 

 that x 3 +a = y 2 is impossible if a is of the form c 3 4 a 6 2 , where b and c are odd, 

 while b has no divisor 4Z+3, and c= 1, 3, 7 (mod 8) if = 1, c=3, 5, 7 (mod 

 8) if a>l. Also if a = 8(2d+l) 3 6 2 , and 6 is prime to 3 and does not have 

 two factors (equal or distinct) of the form 4Z+3; for example, a= 17 or 

 47. Also, if a = 8c 3 26 2 , where c = 4&+ 1 and 6 is an odd number not having 

 two equal or distinct prime factors of the forms 8Z+5 or 8Z+7; for example, 

 a = 6, - - 10, 1 18, - 58. Also, if a = Sc 3 + 26 2 , c = 4fc + 3, and b is odd and with- 

 out two prime factors SZ+3 or 8Z+5. Also in several analogous cases. 



E. Catalan 16 noted that some, but not all, solutions of x 2 +3y 2 = z* are 



Z = K +)(- 2/3) (0-2a), 2/ = fa/3(a-/3), z = a 2 - a p+P 2 . 

 S. Realis 17 gave identities showing solutions of x 3 -\-k = y 2 if k = b 2 (8b 3a 2 ), 

 6 2 (6-3a 2 ), 6(3a 2 +6) 2 , 4a 2 (a 2 +l). Given one solution a 3 +k = f3 2 , another 

 follows from the identity, obtained by Euler's 5 process, 



/27a 6 -36a 3 /3 2 +8/3 4 V 



V 



J 



Realis 18 stated that, if z 2 3az a 3 +/3 2 = has integral roots, 



has integral solutions x = a z, y = (3z; for example, if a = a 2 , |3= 

 a = 2, /3 = 1; a = 32, |8 = 64. If 



a; 3 a 3 +/3 2 = 2/ 2 has integral solutions other than x = a, y = p. Cf. Ch. XXI. 346 

 T. Pepin 19 proved there is one and only one square which becomes an 

 odd [Pepin 33 ] cube on adding 2, 13, 47, 49, 74, 121, 146, 191, 193, 301, 506, 

 589, 767, 769, 866 or 868. No square >0 added to 1, 3, 5, 27, 50, 171, or 

 475 becomes an odd cube. The only solutions of x 2 -\-ll = y 3 are x = 4, 58; 

 the only solution of x 2 +19 = y 3 is y = 7. If a is one of the primes 11, 17, 

 29, 37, 47, 83, 96, 107, 181, 197, 233, 359, 421, 569, 757, 827, there is a 

 single square which becomes an odd cube on adding lla 2 . If a < 1000 and 

 a is of one of the linear forms 38Z+3, 13, 15, 21, 27, 29, 31, 33, 37 and a + 29, 

 89, 173, 281, 331, 569, 953, no square increased by 19a 2 is an odd cube. Also, 

 similar theorems. 



Pepin 20 gave sixteen special theorems on x-+g = z z , proved only under 

 the assumption that x is even and z odd. 



Pepin 21 proved that x^n^z 3 if n = 5, 6, 10, 12, 14, -, 98; 4z 2 +n=M 3 

 if n = 7, 15, 39, 47, 55, 63, 71, 79; z 2 +44 = s 3 only for a; 2 = 81; and gave 

 several theorems on x 2 -\-lly 2 = z z [all provided z is odd, Pepin 33 ]. 



15 Annales Soc. Sc. Bruxelles, 6, 1881-2, 86-100. 

 Mem. Soc. Sc. Liege, (2), 10, 1883, No. 1, p. 10. 



17 Nouv. Ann. Math., (3), 2 ,1883, 289-297. 



18 Ibid., 334-5. Proof of first by E. Fauquembergue, (3), 4, 1885, 379; of second by H. 



Brocard, (3), 10, 1891, p. 7* of Exercices. 



19 Mem. Pont. Accad. Nuovi Lincei, 8, 1892, 41-72; Extract, Sphinx-Oedipe, 1908-9, 188-9. 



Cf. Pepin. 75 



20 Comptes Rendus Paris, 119, 1894, 397-9; corrections, 120, 1895, 494 [Pepin 33 ]. 

 21 Ibid., 120, 1895, 1254-6. 



