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A081120
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Number of integral solutions to Mordell's equation y^2 = x^3 - n.
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24
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1, 2, 0, 4, 0, 0, 4, 1, 0, 0, 4, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 0, 2, 0, 2, 4, 1, 6, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 2, 0, 0, 0, 2, 2, 0, 6, 4, 2, 0, 0, 0, 4, 2, 4, 2, 0, 0, 0, 4, 2, 0, 4, 1, 0, 0, 2, 0, 0, 0, 2, 2, 0, 2, 0, 4, 0, 0, 2, 0, 2, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 6
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OFFSET
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1,2
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COMMENTS
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Mordell's equation has a finite number of integral solutions for all nonzero n.
Gebel, Petho, and Zimmer (1998) computed the solutions for |n| <= 10^4. Bennett and Ghadermarzi (2015) extended this bound to |n| <= 10^7.
Sequence A081121 gives n for which there are no integral solutions. See A081119 for the number of integral solutions to y^2 = x^3 + n.
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, page 191.
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LINKS
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Jean-François Alcover, Table of n, a(n) for n = 1..10000 [There were errors in the previous b-file, which had 10000 terms contributed by T. D. Noe and based on the work of J. Gebel.]
Bennett M.A; Ghadermarzi A. Mordell's equation : a classical approach. LMS J. Compute. Math. 18 (2015): 633-646. doi:10.1112/S1461157015000182 arXiv:1311.7077
Gebel, J.; Pethö, A.; Zimmer, H. G. On Mordell's equation, Compositio Mathematica. 110:3 (1998): 335-367.
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, G. Zimmer, On Mordell's equation, Compositio Mathematica 110 (1998), 335-367.
Eric Weisstein's World of Mathematics, Mordell Curve
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MATHEMATICA
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(* This naive approach gives correct results up to n=1000 *) xmax[_] = 10^4; Do[ xmax[n] = 10^5, {n, {366, 775, 999}}]; Do[ xmax[n] = 10^6, {n, {207, 307, 847}}]; f[n_] := (x = Floor[n^(1/3)] - 1; s = {}; While[ x <= xmax[n], x++; y2 = x^3 - n; If[y2 >= 0, y = Sqrt[y2]; If[ IntegerQ[y], AppendTo[s, y]]]]; s); a[n_] := (fn = f[n]; If[fn == {}, 0, 2 Length[fn] - If[ First[fn] == 0, 1, 0]]); Table[ an = a[n]; Print["a[", n, "] = ", an]; an, {n, 1, 100}] (* Jean-François Alcover, Mar 06 2012 *)
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CROSSREFS
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Cf. A081119, A081121.
Sequence in context: A338620 A258701 A246160 * A200038 A249093 A102392
Adjacent sequences: A081117 A081118 A081119 * A081121 A081122 A081123
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KEYWORD
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nice,nonn
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AUTHOR
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T. D. Noe, Mar 06 2003
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EXTENSIONS
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Edited by Max Alekseyev, Feb 06 2021
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STATUS
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approved
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