A103254 Positive integers x such that there exist positive integers y and z satisfying x^3 + y^3 = z^2.

1, 2, 4, 7, 8, 9, 10, 11, 14, 16, 18, 21, 22, 23, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 44, 46, 49, 50, 56, 57, 63, 64, 65, 70, 72, 78, 81, 84, 86, 88, 90, 91, 92, 95, 98, 99, 100, 104, 105, 110, 112, 114, 121, 122, 126, 128, 129, 130, 132, 136, 140, 144, 148, 152, 154, 158, 160, 162, 169, 170, 175, 176, 177, 183, 184, 189, 190, 193, 196, 198, 200

Offset: 1

Author

Cino Hilliard, Mar 20 2005

Keywords

• nonn

Comments

A001105 is a subset (excluding 0), since (x, y, z) = (A001105(k), A001105(k), A033430(k)) satisfies x^3 + y^3 = z^2. - R. J. Mathar, Sep 11 2006

A parametric solution: {x,y,z} = {g*(4*e + g)*(4*e^2 + 8*e*g + g^2), 2*g*(4*e + g)*(-2*e^2 +2*e*g + g^2), 3*g^2*(4*e + g)^2*(4*e^2 + 2*e*g + g^2)}, provided (-2*e^2 +2*e*g + g^2) > 0. - James Mc Laughlin, Jan 27 2007

Allowing y = 0 would give the same sequence, since x^3 = z^2 implies x is a square, and all squares are terms since (t^2)^3 + (2*t^2)^3 = (3*t^3)^2. On the other hand, allowing y to be negative would introduce new terms: 71, 74, and 155 would be terms since 71^3 + (-23)^3 = 588^2, 74^3 + (-47)^3 = 549^2, and 155^3 + (-31)^3 = 1922^2. See A356720. - Jianing Song, Aug 24 2022

Examples

			x=7, y=21, 7^3 + 21^3 = 98^2. 7 is the 4th term in the list.
Other solutions are (x, y, z)=(1, 2, 3), (4, 8, 24), (7, 21, 98), (9, 18, 81), (10, 65, 525), (11, 37, 228), (14, 70, 588), (16, 32, 192), (21, 7, 98), (22, 26, 168), (23, 1177, 40380), ...
		

Links

• Fritz Beukers, The Diophantine equation Ax^p+By^q=Cz^r, Duke Math. J. 91 (1998), 61-88.

Crossrefs

See A103255 for another version.

Programs

Magma

[ k : k in [1..200] | exists{P : P in IntegralPoints(EllipticCurve([0,k^3])) | P[1] gt 0 and P[2] ne 0 } ]; // Geoff Bailey, Jan 28 2007

Extensions

Recomputed and extended to 48 terms by Geoff Bailey (geoff(AT)maths.usyd.edu.au) using Magma, Jan 28 2007

Terms 104..200 added by Joerg Arndt, Sep 29 2012