A020897 Sum of two nonzero rational cubes.

2, 6, 7, 9, 12, 13, 15, 16, 17, 19, 20, 22, 26, 28, 30, 31, 33, 34, 35, 37, 42, 43, 48, 49, 50, 51, 53, 54, 56, 58, 61, 62, 63, 65, 67, 68, 69, 70, 71, 72, 75, 78, 79, 84, 85, 86, 87, 89, 90, 91, 92, 94, 96, 97, 98, 103, 104, 105, 106, 107, 110, 114, 115, 117, 120

Offset: 1

Author

Steven Finch

Keywords

• nonn
• nice

Comments

n such that x^3 + y^3 = n*z^3 has a solution in nonnegative integers x,y,z.

Dolan on page 37 states: "Equation 1 is insoluble if n = 31, 38, 67, 76 or 95." - Michael Somos, Nov 18 2021

See the Selmer 1951 article pp. 357-360, Table 6, "The number g of generators and the basic solutions of the equation X^3 + Y^3 = AZ^3, A cubefree and <= 500. - Michael Somos, Feb 15 2022

Examples

			6*21^3 = 37^3 + 17^3, 7*3^3 = 5^3 + 4^3, 9*1^3 = 2^3 + 1^3, 12*39^3 = 89^3 + 19^3, 13*3^3 = 7^3 + 2^3, 15*294^3 = 683^3 + 397^3, ... - _Michael Somos_, Nov 18 2021
31*42^3 = 137^3 + (-65)^3, 67*1323^3 = 5353^3 + 1208^3. - _Michael Somos_, Feb 12 2022
		

Links

• S. W. Dolan, On expressing numbers as the sum of two cubes, The Mathematical Gazette, Vol. 66, No. 436 (Mar., 1982), pp. 31-38. See table of solutions on page 32. - _Michael Somos_, Nov 18 2021
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]
• Ernst S. Selmer, The diophantine equation ax^3 + by^3 + cz^3 = 0, Acta Math. 85 (1951), pp. 203-362.

Crossrefs

Subsequence of A159843.

Programs

Mathematica

(* A naive program with a few pre-computed terms *) nmax = 120; xmax = 2000; CubeFreePart[n_] := Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 3]} & /@ FactorInteger[n]); nn = Reap[Do[n = CubeFreePart[x*y*(x+y)]; If[1 < n <= nmax, Sow[n]], {x, 1, xmax}, {y, x, xmax}]][[2, 1]] // Union; A020897 = Select[ Union[nn, nn*2^3, nn*3^3, nn*4^3, {17, 31, 53, 67, 71, 79, 89, 94, 97, 103, 107}], # <= nmax &] (* Jean-François Alcover, Apr 02 2012 *)

Extensions

Offset corrected by Arkadiusz Wesolowski, Aug 15 2013