CHAP, xxi] NUMBERS THE SUM OF Two RATIONAL CUBES. 575 



17, 38, and all solutions with z even. This solution is simpler than 

 Fermat's 41 1265, --1256, 183. 



S. Ralis 197 noted that, from the solution 1, 2, 1 of a; 3 -H/ = 9z 3 , Prestet's 

 formulas give the solution 17, -20, -7, from which the new formulas 



give 3-919, -3-271, 3-438 and hence the solution by Lucas. 195 For A = 7, 

 an analogous second set of formulas was given by Realis. 



Lucas 198 noted that integral solutions exist if and only if A is of the form 

 ab(a+6)/c 3 , where a, b, c are integers. For, if x, y, z are solutions, a = x 3 , 

 b = y* give ab(a-\-b) = A(xyz) 3 . The converse is true by the identity 



For x = l, y = 2, we get 17 3 +37 3 = 6-21 3 , contrary to Legendre. 184 



Lucas 199 proved Sylvester's theorem that the equation is impossible 

 for A=p, 2p, 4<?, <? 2 , 4p 2 , 2q 2 , where p and q are primes 18Z+5, 18Z+11, 

 respectively. Combining this result with that of Lucas, 198 we see that 

 xy(x-\-y) = Az* is impossible in rational numbers (excluding zero and equal 

 values) if A = p, 2p, 4q, 4p 2 , q\ 2q 2 , 1, 2, 3, 4, 18, 36. 



A. Desboves 200 derived the identity (2) by Lucas from Lagrange's 163 

 theory of polynomials which repeat under multiplication. 



J. J. Sylvester 201 proved that pq, p 2 g 2 , ppl, qql are not sums of two rational 

 cubes if p, pi are primes 18Z+5 and q, qi primes 18Z+11. These with 

 P) ?> P 2 > 2 2 > tne i r products by 9, and 2p, 4q, 4p 2 , 2g 2 , give all known types 

 not resolvable into a sum or difference of two rational cubes. He an- 

 nounced the theorem that if p, ^, are primes of the respective forms 

 18n+l, + 7, + 13, while each is not of the form/ 2 +270 2 and hence does 

 not have 2 as a cubic residue, then no one of the numbers 2p, 4p, 2p 2 , 4p 2 , 

 2^, 4^ 2 , 40, 20 2 is a sum of two rational cubes. If v is a prime 6n+l not 

 having 3 as a cubic residue, then neither 3v nor 3v 2 is a sum of two cubes. 

 By all of these results, we know whether or not any number ^ 100 (except 

 perhaps 66) is a sum of two rational cubes. Proofs of the above theorems 

 rest on the linear form of the divisors of x 3 3x+l. He stated the em- 

 pirical theorem that every prime 18nl or else its triple is expressible in 

 the form 202 x 3 -3xy 2 y*. 



A. Desboves 203 gave two proofs of Lucas' identity (2) and noted that 

 the replacement of x by x 3 and y by y 3 yields Lucas' (1). He showed that 



197 Nouv. Ann. Math., (2), 17, 1878, 454-7. 



198 Ibid., 425-6. Cf . Candido 179 of Ch. XXIII. 



199 Ibid., 507-14. This and his 193 preceding paper are duplicated in Amer. Jour. Math., 2, 



1879, 182-4. 



200 Comptes Rendus Paris, 87, 1878, 159. 



201 Comptes Rendus Paris, 90, 1880, 289, 1105 (correction); Amer. Jour. Math., 2, 1879, 280, 



389-393. Coll. Math. Papers, 3, 1909, 430, 437; 312, 347-9. 



202 A. M. Sawin, Annals of Math., 1, 1884-5, 58-63, noted that x and y are relatively prime 



integers if and only if n is an integer. 



203 Nouv. Ann. Math., (2), 18, 1879, 400, 491; (3), 5, 1886, 577. 



