CHAP, xxi] THREE EQUAL SUMS OF Two CUBES. 561 



manuscript by Baba, solved (10) by setting 



x = a+b, y = a-b, z = bc, u = d-bc, 2a?+6ab 2 -d 3 +3bcd--3b' 2 c 2 d = Q. 



Take a = c 2 d/2. Then 126c = d(4-c 6 ). Take c 3 = , a 2 -4 = /3, and multiply 

 the resulting values of x, y, z, u by 12cfd; we get 



z=-c u 



M. Weill 92 noted that if a;,-, ?/;, 2,-, w,- give two solutions of (10), we can 

 evidently find 5 rationally so that Xi-\-5x 2 , , Wi + 5w 2 is a solution. 

 Given only one solution, we obtain a new solution x\-\-pt, y \-\-\t, 

 t, if AP+3Bt+3C = 0, where 



We may choose X, , p to make (7 = or A = and get t rationally. 



For three consecutive cubes whose sum is a cube, see papers 245-267. 

 For minor results on our subject, see Schier 67 of Ch. XXIII. 



THREE EQUAL SUMS OF TWO CUBES. 



Fermat's 40 method of solution was given above. 



W. Lenhart 93 found four integers the sum of any two of which is a cube. 

 Three of the conditions are satisfied if x, m?x, n 3 x, r*x be taken as 

 the numbers. The remaining conditions require that ra 3 +n 3 2x, w 3 +r 3 

 2x, n 3 +r 3 2x be cubes, say s 3 , a 3 , 6 3 . Eliminating x, we have 



(1) r 3 +s 3 = a 3 +n 3 = & 3 +w 3 . 



By his 186 table of numbers expressible as a sum of two cubes, 

 46969 = ( W+ C^) 3 = (W) 3 + (W) 3 = ( 3 &fff 



Rejecting the common denominator, we get integers (one of 24 digits and 

 three of 22 digits) solving the initial problem. 



A. B. Evans 94 obtained the last result otherwise. By Euler, 51 for 



Now 13-3613 = 41 3 28 3 can be expressed as a sum of two cubes by the 

 usual method. The final answer involves numbers of 22 and 24 digits. 



J. Matteson 95 obtained Lenhart's result by the method of Evans. 



H. Brocard 96 noted that the sum of any two of the numbers 20012^, 

 -15916|, 19291|, -20020| is a cube. E. B. Escott 97 noted that 6044, 

 7780, -1948, -6052 have this property. 



E. Fauquembergue 98 gave an erroneous solution of (1) with 5 parameters. 



92 Nouv. Ann. Math., (4), 17, 1917, 41-46. 



93 Math. Miscellany, New York, 1, 1836, 155-6. 



94 Math. Quest. Educ. Times, 15, 1871, 91-2. His factor 2 3 should be 2 6 . 



96 Collection Dioph. Problems, pub. by A. Martin, Washington, D. C., 1888, 1-4. 



96 L'interme'diaire des math., 8, 1901, 183-4. 



97 Ibid., 9, 1902, 16. 



"Ibid., 9, 1902, 155; 10, 1903, 82 (Sphinx-Oedipe, 1906-7, 80, 125). 

 37 



