

A029728


Complete list of solutions to y^2 = x^3 + 17; sequence gives x values.


17




OFFSET

1,1


COMMENTS

Comments by Henri Cohen on the proof that the list of solutions is complete: (Start)
This is now completely standard. Cremona's mwrank program tells us that this is an elliptic curve of rank 2 with generators P1=(2,3) and P2=(4,9).
We now apply the algorithm (essentially due to Tzanakis and de Weger) described in Nigel Smart's book on the algorithmic solution of Diophantine equations: using Sinnou David's bounds on linear forms in elliptic logarithms one finds that if P is an integral point then P=aP1+bP2 for a and b less than a huge bound B (typically 10^100, sometimes more, I didn't do the computation here), but the main point is that B is completely explicit. One then uses the LLL algorithm: this is crucial.
A first application reduces the bound to 200, say, then a second application to 20 and sometimes a third to 12 (at this point it is not necessary). Then a very small search gives all the possible integer points. (End)


REFERENCES

L. J. Mordell, Diophantine Equations, Ac. Press, p. 246.
T. Nagell, Einige Gleichungen von der Form ay^2+by+c=dx^3, Vid. Akad. Skrifter Oslo, Nr. 7 (1930).
Silverman, Joseph H. and John Tate, Rational Points on Elliptic Curves. New York: SpringerVerlag, 1992.


LINKS



MATHEMATICA

ok[x_] := Reduce[y>0 && y^2 == x^3 + 17, y, Integers] =!= False; Select[Table[x, {x, 2, 10000}], ok ] (* JeanFrançois Alcover, Sep 07 2011 *)


PROG

(Magma) Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 17])) ]); // Sergei Haller (sergei(AT)sergeihaller.de), Dec 21 2006
(SageMath) [i[0] for i in EllipticCurve([0, 17]).integral_points()] # Seiichi Manyama, Aug 25 2019


CROSSREFS

x values of solutions to y^2 = x^3 + a*x + b;


KEYWORD

sign,fini,full


AUTHOR



STATUS

approved



