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Positive numbers of the form y^2 - x^3, x and y >= 1.
6

%I #34 Mar 13 2023 10:21:59

%S 1,3,8,9,12,15,17,18,19,22,24,28,30,35,36,37,38,40,41,44,48,54,55,56,

%T 57,63,64,65,68,71,73,79,80,89,92,94,97,98,99,100,101,105,106,107,108,

%U 112,113,117,119,120,121,128,129,131,132,136,138,141,142,143,145,148,151

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

%C From _Artur Jasinski_, Oct 03 2007: (Start)

%C Some numbers have multiple partitions:

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

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

%C 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

%H J. Gebel, <a href="/A001014/a001014.txt">Integer points on Mordell curves</a> [Cached copy, after the original web site tnt.math.se.tmu.ac.jp was shut down in 2017]

%H J. Gebel, A. Petho and H. G. Zimmer, <a href="http://dx.doi.org/10.1023/A:1000281602647">On Mordell's equation</a>, Compositio Math. 110 (1998), 335-367.

%H Cino Hilliard, <a href="http://groups.msn.com/BC2LCC/n37ltk2.msnw">Proof that n^3+7 <> k^2 for all integers n,k</a>.

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

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

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

%Y Complement of A080762.

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

%Y Cf. A029727, A029728, A134042, A134043.

%K nonn

%O 1,2

%A _Cino Hilliard_, Mar 10 2003

%E "Positive" added to definition by _N. J. A. Sloane_, Oct 06 2007