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Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.
5

%I #12 Feb 16 2025 08:33:06

%S 1,1,0,2,0,0,2,1,0,0,2,0,1,0,1,0,0,1,1,1,0,0,1,0,1,2,1,3,0,0,0,0,0,0,

%T 1,0,0,0,3,1,0,0,0,1,1,0,3,2,1,0,0,0,2,1,2,1,0,0,0,2,1,0,2,1,0,0,1,0,

%U 0,0,1,1,0,1,0,2,0,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,0,3,0,0,0,3

%N Number of integral solutions with nonnegative y to Mordell's equation y^2 = x^3 - n.

%C a(n) = A081120(n)/2 if A081120(n) is even, (A081120(n)+1)/2 if A081120(n) is odd (i.e. if n is a cubic number).

%C Comment from T. D. Noe, Oct 12 2007: In sequences A134108 and A134109 (this entry) dealing with the equation y^2 = x^3 + n, one could note that these are Mordell equations. Here are some related sequences: A054504, A081119, A081120, A081121. The link "Integer points on Mordell curves" has data on 20000 values of n. A134108 and A134109 count only solutions with y >= 0 and can be derived from A081119 and A081120.

%H Jean-François Alcover, <a href="/A134109/b134109.txt">Table of n, a(n) for n = 1..10000</a>

%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 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MordellCurve.html">Mordell Curve</a>

%e y^2 = x^3 - 4 has solutions (y, x) = (2, 2) and (11, 5), hence a(4) = 2.

%e y^2 = x^3 - 5 has no solutions, hence a(5) = 0.

%e y^2 = x^3 - 8 has solution (y, x) = (0, 2), hence a(8) = 1.

%e y^2 = x^3 - 207 has 7 solutions (see A134106, A134107), hence a(207) = 7.

%t A081120 = Cases[Import["https://oeis.org/A081120/b081120.txt", "Table"], {_, _}][[All, 2]];

%t a[n_] := With[{an = A081120[[n]]}, If[EvenQ[an], an/2, (an+1)/2]];

%t a /@ Range[10000] (* _Jean-François Alcover_, Nov 28 2019 *)

%o (Magma) [ #{ Abs(p[2]) : p in IntegralPoints(EllipticCurve([0, -n])) }: n in [1..104] ];

%Y Cf. A081120, A134106, A134107, A134108.

%K nonn,changed

%O 1,4

%A _Klaus Brockhaus_, Oct 08 2007, Oct 14 2007