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Mordell Curve

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A Mordell curve is an elliptic curve of the form y^2=x^3+n for n a nonzero integer. This equation has a finite number of solutions in integers. If (x,y) is a solution, it therefore follows that (x,-y) is as well.

MordellCurve

Uspensky and Heaslet (1939) give elementary solutions for n=-4, -2, and 2, and then give n=-1, -5, -6, and 1 as exercises. Euler found that the only integer solutions to the particular case n=1 (a special case of Catalan's conjecture) are (x,y)=(-1,0), (0,+/-1), and (2,+/-3). This can be proved using Skolem's method, using the Thue equation x^3-2y^3=+/-1, using 2-descent to show that the elliptic curve has rank 0, and so on. It is given as exercise 6b in Uspensky and Heaslet (1939, p. 413), and proofs published by Wakulicz (1957), Mordell (1969, p. 126), Sierpiński and Schinzel (1988, pp. 75-80), and Metsaenkylae (2003).

Solutions of the Mordell curve with 0<=y<10^5 are summarized in the table below for small n.

nsolutions
1(-1,0),(0,1),(2,3)
2(-1,1)
3(1,2)
4(0,2)
5(-1,2)
6none
7none
8(-2,0),(1,3),(2,4),(46,312)
9(-2,1),(0,3),(3,6),(6,15),(40,253)
10(-1,3)

Values of n such that the Mordell curve has no integer solutions are given by 6, 7, 11, 13, 14, 20, 21, 23, 29, 32, 34, 39, 42, ... (OEIS A054504; Apostol 1976, p. 192).

See also

Catalan's Conjecture, Catalan's Diophantine Problem, Elliptic Curve

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References

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.Cohen, H. "y^2=x^3+1." 24 Nov 2003. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0311&L=nmbrthry&F=&S=&P=1197.Conrad, M. Untitled. http://emmy.math.uni-sb.de/~simath/MORDELL/MORDELL+.Gebel, J. "Data on Mordell's Curve." http://tnt.math.metro-u.ac.jp/simath/MORDELL/.Gebel, J.; Pethő, A.; and Zimmer, H. G. "On Mordell's Equation." Compos. Math. 110, 335-367, 1998.Llorente, P. and Quer, J. "On the 3-Sylow Subgroup of the Class Group of Quadratic Fields." Math. Comput. 50, 321-333, 1988.Mestre, J.-F. "Rang de courbes elliptiques d'invariant donné." C.R. Acad. Sci. Paris 314, 919-922, 1992.Mestre, J.-F. "Rang de courbes elliptiques d'invariant nul." C.R. Acad. Sci. Paris 321, 1235-1236, 1995.Metsaenkylae, T. "Catalan's Conjecture: Another Old Diophantine Problem Solved." Bull. Amer. Math. Soc. S 0273-0979(03)00993-5, September 5, 2003.Mordell, L. J. Diophantine Equations. London, England: Academic Press, 1969.Myerson, G. "Re: y^2=x^3+1." 24 Nov 2003. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0311&L=nmbrthry&F=&S=&P=1290.Quer, J. "Corps quadratiques de 3-rang 6 et courbes elliptiques de rang 12." C.R. Acad. Sci. Paris. Sér. 1 Math. 305, 215-218, 1987.Sierpiński, W. and Schinzel, A. Elementary Theory of Numbers, 2nd Eng. ed. Amsterdam, Netherlands: North-Holland, 1988.Sloane, N. J. A. Sequence A054504 in "The On-Line Encyclopedia of Integer Sequences."Szymiczek, K. "Re: y^2=x^3+1." 26 Nov 2003. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0311&L=nmbrthry&F=&S=&P=1492.Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, 1939.Wakulicz, A. "On the Equation x^3+y^3=2z^3." Colloq. Math. 5, 11-15, 1957.Womack, T. "Minimal-Known Positive and Negative k for Mordell Curves of Given Rank." http://www.maths.nott.ac.uk/personal/pmxtow/mordellc.htm.

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Mordell Curve

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Weisstein, Eric W. "Mordell Curve." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/MordellCurve.html

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