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



x 3 +?/ 3 = Az 3 has integral solutions if A = xy(x-\-y}, 3 +?/ 3 , 2x 6 -\-Qy~, x(y 3 x}, 

 or x*-y*-3xy(x+2y}, and hence if A = 6, 7, 9, 12, 15, 17, 19, 20, 22, 26, 

 28, 30, 37. 



E. Catalan 204 noted that xy(x-\-y) =z z is impossible in view of the 

 identity (2) and the impossibility of r 3 -f s 3 = 3 . Lucas' 198 paper implies this 

 result. 



E. Lucas 205 proved certain and stated others of the preceding theorems 

 by Sylvester 201 and Pepin, 193 and remarked that, if .T 3 3xy-+y* = 3Az 3 has 

 solutions, then 206 



the divisors of the resulting A's being of the form 18wl. In the third 

 paper he cited cases (A a prime 18/1+13, A a square of a prime 18n+7, etc.) 

 in which x 3 +y 3 = Az 3 can be completely solved by the method of tangents 

 and secants, citing Sylvester's theory of residuation. 



T. Pepin 207 proved (p. 110) Sylvester's 201 theorem on 2p, 4p, 2^, etc., 

 and remarked (p. 75) that the first three are covered by the method used 

 by Pepin 194 for 2-7, 2-19, 4-19. He proved (p. 109) the results stated by 

 Sylvester 201 on the 16 types pq, , 2g 2 , as well as the theorem (pp. 113-4) : 

 If 



p = (9m+4) 2 +3(9/i4) 2 , ^= (9ra+2) 2 +3(9w2) 2 , 



are primes, no one of the numbers 



i8( P , t, r, < 2 , **, r 2 ), 36( P , 0, r, P 2 , y, t 2 ) 



is a sum of two rational cubes. 



C. Henry 208 proved that any number of the form A=f 1 - 9# 12 and its 

 double are expressible as sums of two cubes : 



= r-c 



if B=f z -g, C=/ 12 +3^ 12 . 



H. Delannoy 209 proved by descent that 3 +?/ 3 = 42 3 is impossible. 

 The problem x* +7/ 3 = 20 3 - 105489 has been treated. 210 

 T. R. Bendz 211 misquoted Lucas' (2), whence his criticism is invalid. 

 K. Schwering 212 put the equation into the form 



204 Nouv. Corresp. Math., 5, 1879, 91. 



206 Bull. Soc. Math. France, 8, 1879-80, 173-182; Comptes Rendus Paris, 90, 1880, 855-7; 



Nouv. Ann. Math., (2), 19, 1880, 206-11. Related results from these papers are quoted 

 under Lucas 70 of Ch. XXV. 



208 Sylvester, Comptes Rendus Paris, 90, 1SSO, 347 (Coll. Math. Papers, III, 432), had stated 



that there exist solutions in functions of degree 9. 



207 Atti Accad. Pont. Nuovi Lincei, 34, 1880-1, 73-131. 



208 Nouv. Ann. Math., (2), 20, 1881, 418-20. The right member of his formula (3) is A, in 

 error for 2 A . 



209 Jour. math. 616mentaires, (5), 1 (ann<Se 21), 1897, 58-9. 



210 Amer. Math. Monthly, 5, 1898, 181. 



211 Ofver diophantiska ekvationen x n +y n = z n , Diss., Upsala, 1901, 15-18. 



212 Archiv Math. Phys., (3), 2, 1902, 285. 



