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



is a representation of mm'. Taking u' = v, v' = u, we get m 2 =f(2uvu?, 

 2uvv 2 ). First, if x+y is not divisible by 3, it is relatively prime to 

 /= (x+y)* 3xy, so that it and / are cubes. By the above, 



When this is taken as/, the sum u?+v 3 of the arguments is a cube (corre- 

 sponding to x+y). Thus the method of descent applies. The case in 

 which x+y is a multiple of 3 leads by a like argument to a descent. 

 P. Bachmann 33 amplified the proofs by Euler 8 and Legendre. 20 

 R. Fueter 34 proved that if 3 +7? 3 +r 3 = is solvable by numbers 4-0 of 

 an imaginary quadratic domain fc(Vm), where ra<0, m=2 (mod 3), then 

 the class number of k is divisible by 3. It is solvable in the real domain 

 fc(V 3m) if and only if solvable_in &(Vw). In particular, Kummer's 

 result that it is not solvable in k( V^3) is a consequence of the fact that it 

 is not solvable in rational numbers. To give a direct proof, let c* 3 +/3 3 = 2 3 , 

 ^3), where x, y, z are integers distinct from 0, and set a 3 , 

 3 2 3 = X. Then 



X+Y\*X-Y 



If m and n are integers prime to 3, the domain defined by a cube root of 

 w*+27n 3 has its class number a multiple of 3, and Z 3 = is solvable. 



W. Burnside 35 discussed the solution of x z +y*+z* = Q in quadratic 

 domains. 



R. D. Carmichael 36 gave a series of lemmas leading to a proof of the fact, 

 stated by Euler, 8 that p 2 +3q 2 = s z (p, q relatively prime, s odd) implies that 

 s is of the form Z 2 +3tt 2 , etc. 



Further proofs by Holden 80 ; also Korneck, 149 Stockhaus 231 , and Rychlik 232 

 of Ch. XXVI. 



Two EQUAL SUMS OF Two CUBES. 



Diophantus, V, 19, mentioned without details the theorem in the Porisms 

 that the difference of two cubes is always a sum of two cubes (cf. p. 607). 



P. Bungus 37 remarked that while a square is often the sum of two 

 squares, a cube is first composed of three cubes, citing 6 3 = 3 3 +4 3 -f5 3 . 



F. Vieta 38 required two cubes whose sum equals the difference B 3 D 3 

 of two given cubes (B>D). Call B A the side of the first required cube 

 and B Z A/D 2 -D the side of the second. Thus (B 3 +D*)A = 3D 3 and hence 



3 - 



(1) 



33 Niedere Zahlentheorie, 2, 1910, 454-8. 



84 Sitzungsber. Akad. Wise. Heidelberg (Math.), 4, A, 1913, No. 25. 

 36 Proc. London Math. Soc., (2), 14, 1914, 1. 

 36 Diophantine Analysis, 1915, 67-70. 

 87 Numerorum Mysteria, 1591, 1618, 463; Pars Altera, 65. 



"Zetetica, 1591, IV, 18-20; Opera Mathematica, ed. by Frans van Schooten, Lugd. Batav., 

 1646, 74-75. A wrong sign in (2) is corrected on p. 554. 



