A338933 Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.
2, 16, 128, 1024, 1458, 8192, 11664, 31250, 65536, 93312, 235298, 524288, 746496, 1062882, 2000000, 3543122, 3906250, 5971968, 9653618, 15059072, 22781250, 28697814, 33554432, 47775744, 48275138, 68024448, 80707214, 94091762, 128000000, 171532242, 226759808
Offset: 1
Keywords
Examples
16 is a term, because when t is an integer, (6*(2*t + 1) - 9*(2*t + 1)^4, 9*(2*t + 1)^4, 2 - 9*(2*t + 1)^3) is a nontrivial primitive parametric solution of x^3 + y^3 + 2*z^3 = 16.
References
- R. K. Guy, Unsolved Problems in Number Theory, D5.
Links
- Kenji Koyama, On searching for solutions of the Diophantine equation x^3 + y^3 + 2z^3 = n, Math. Comp, 69 (2000), 1735-1742.
- J. C. P. Miller & M. F. C. Woollett, Solutions of the Diophantine equation x^3 + y^3 + z^3 = k, J. London Math. Soc. 30(1955), 101-110.
- Beniamino Segre, On the rational solutions of homogeneous cubic equations in four variables, Math. Notae, 11 (1951), 1-68.
Programs
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Mathematica
t1 = 2*Range[23]^6; t2 = 2*{1, 2, 4, 5, 7, 8}^9; t3 = 1458*Range[4]^9; t4 = 2*{1, 5}^12; t5 = 16*{1, 2, 4}^12; t6 = 1458*{1, 3}^12; t7 = 11664*{1, 2, 3}^12; Take[Union[t1, t2, t3, t4, t5, t6, t7], 31]
Extensions
Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022
Comments