A080761 Positive numbers of the form y^2 - x^3, x and y >= 1.

1, 3, 8, 9, 12, 15, 17, 18, 19, 22, 24, 28, 30, 35, 36, 37, 38, 40, 41, 44, 48, 54, 55, 56, 57, 63, 64, 65, 68, 71, 73, 79, 80, 89, 92, 94, 97, 98, 99, 100, 101, 105, 106, 107, 108, 112, 113, 117, 119, 120, 121, 128, 129, 131, 132, 136, 138, 141, 142, 143, 145, 148, 151

Offset: 1

Author

Cino Hilliard, Mar 10 2003

Keywords

• nonn

Comments

From Artur Jasinski, Oct 03 2007: (Start)

Some numbers have multiple partitions:

8 = 4^2 - 8^3 = 312^2 - 46^3,

9 = 6^2 - 3^3 = 15^2 - 6 ^3 = 253^2 - 40^3. (End)

This is Mordell's equation with the condition that x and y are positive. Sequence A054504 lists the n for which there is no solution to Mordell's equation. Hence, none of those numbers will be in this sequence. The terms of this sequence can be determined by looking at the link to Gebel's data. - T. D. Noe, Mar 23 2011

Examples

			8 is in the sequence since 3^2 = 1^3 + 8.
		

Links

• J. Gebel, Integer points on Mordell curves [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]
• J. Gebel, A. Petho and H. G. Zimmer, On Mordell's equation, Compositio Math. 110 (1998), 335-367.
• Cino Hilliard, Proof that n^3+7 <> k^2 for all integers n,k.

Crossrefs

Complement of A080762.

Cf. sequences for n^3+7, n^3+17, n^3+3, n^3+2, n^3+5.

Cf. A029727, A029728, A134042, A134043.

Programs

Mathematica

With[{nn=100},Take[Union[Select[First[#]^2-Last[#]^3&/@Tuples[Range[ 20nn],2],#>0&]],nn]] (* Harvey P. Dale, Jul 10 2012 *)

PARI

diop(n,m) = { for(p=1,m, for(x=1,n, y=x*x*x+p; if(issquare(y),print1(p" "); break) ) ) }

Extensions

"Positive" added to definition by N. J. A. Sloane, Oct 06 2007