A266212 Positive integers x such that x^3 = y^4 + z^2 for some positive integers y and z.

8, 13, 20, 25, 40, 125, 128, 193, 200, 208, 225, 313, 320, 328, 400, 500, 605, 640, 648, 1000, 1053, 1156, 1521, 1620, 1625, 1681, 1700, 2000, 2025, 2048, 2125, 2465, 2493, 2873, 2920, 3025, 3088, 3185, 3200, 3240, 3328, 3400, 3600, 3656, 3748, 3816, 4225, 4625, 4913, 5000, 5008, 5120, 5248, 6400, 6728, 6760, 6793, 6845, 7225, 8000

Offset

1,1

Comments

If x^3 = y^4 + z^2, then (a^(4k)*x)^3 = (a^(3k)*y)^4 + (a^(6k)*z)^2 for all a = 1,2,3,... and k = 0,1,2,... So the sequence has infinitely many terms.

Conjecture: For any integer m, there are infinitely many triples (x,y,z) of positive integers with x^4 - y^3 + z^2 = m.

This is stronger than the conjecture in A266152.

Links

Zhi-Wei Sun and Chai Wah Wu, Table of n, a(n) for n = 1..698 n = 1..100 from Zhi-Wei Sun

Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.

Example

a(1) = 8 since 8^3 = 4^4 + 16^2.
a(2) = 13 since 13^3 = 3^4 + 46^2.
a(3) = 20 since 20^3 = 4^4 + 88^2.
a(8) = 193 since 193^3 = 6^4 + 2681^2.
a(12) = 313 since 313^3 = 66^4 + 3419^2.
a(20) = 1000 since 1000^3 = 100^4 + 30000^2.

Mathematica

SQ[n_]:=SQ[n]=n>0&&IntegerQ[Sqrt[n]]
n=0; Do[Do[If[SQ[x^3-y^4], n=n+1; Print[n, " ", x]; Goto[aa]], {y, 1, x^(3/4)}]; Label[aa]; Continue, {x, 1, 8000}]

Crossrefs

Cf. A000290, A000578, A000583, A262827, A266003, A266004, A266152, A266153.

Sequence in context: A219721 A176209 A227453 * A063849 A273980 A101642

Adjacent sequences: A266209 A266210 A266211 * A266213 A266214 A266215

Keyword

nonn

Author

Zhi-Wei Sun, Dec 23 2015

Status

approved