login
A338933
Numbers k such that the Diophantine equation x^3 + y^3 + 2*z^3 = k has nontrivial primitive parametric solutions.
1
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
1,1
COMMENTS
The data are derived from the following formula:
(a^2 - a*t - t^2)^3 + (a^2 + a*t - t^2)^3 + 2*(t^2)^3 = 2*a^6
(a^3 - 3*t^3)^3 + (a^3 + 3*t^3) + 2*(-3*a*t^2)^3 = 2*a^9;
(9*a^3 - t^3)^3 + (9*a^3 + t^3)^3 + 2*(-3*a*t^2)^3 = 1458*a^9;
(6*a^3*t - 72*t^4)^3 + (72*t^4)^3 + 2*(a^4 - 36*a*t^3)^3 = 2*a^12;
(6*a^3*t - 9*t^4)^3 + (9*t^4)^3 + 2*(2*a^4 - 9*a*t^3)^3 = 16*a^12 = 2*2^3*a^12;
(18*a^3*t - 8*t^4)^3 + (8*t^4)^3 + 2*(9*a^4 - 12*a*t^3)^3 = 1458*a^12 = 2*9^3*a^12;
(18*a^3*t - t^4)^3 + (t^4)^3 + 2*(18*a^4 - 3*a*t^3)^3 = 11664*a^12 = 2*18^3*a^12.
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, D5.
LINKS
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.
EXAMPLE
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.
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]
KEYWORD
nonn
AUTHOR
XU Pingya, Nov 16 2020
EXTENSIONS
Missing terms 1024 and 746496 added by XU Pingya, Mar 14 2022
STATUS
approved