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 A029728 Complete list of solutions to y^2 = x^3 + 17; sequence gives x values. 17
 -2, -1, 2, 4, 8, 43, 52, 5234 (list; graph; refs; listen; history; text; internal format)
 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: Springer-Verlag, 1992. LINKS Table of n, a(n) for n=1..8. MATHEMATICA ok[x_] := Reduce[y>0 && y^2 == x^3 + 17, y, Integers] =!= False; Select[Table[x, {x, -2, 10000}], ok ] (* Jean-François Alcover, Sep 07 2011 *) PROG (Magma) Sort([ p[1] : p in IntegralPoints(EllipticCurve([0, 17])) ]); // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006 (SageMath) [i[0] for i in EllipticCurve([0, 17]).integral_points()] # Seiichi Manyama, Aug 25 2019 CROSSREFS Cf. A029727 (y values). x values of solutions to y^2 = x^3 + a*x + b; A134107 (a= 0, b=-207), A134074 (a= 0, b= 73), A134042 (a= 0, b= 113), A134103 (a= 0, b= 225), A134105 (a= 0, b= 297), A134167 (a= 0, b=1025), A316456 (a=-7, b= 10), A309071 (a=20, b= 0). Sequence in context: A024739 A359689 A024959 * A135547 A146307 A343952 Adjacent sequences: A029725 A029726 A029727 * A029729 A029730 A029731 KEYWORD sign,fini,full AUTHOR N. J. A. Sloane STATUS approved

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Last modified August 13 18:32 EDT 2024. Contains 375144 sequences. (Running on oeis4.)