Geodesic
Integral points on elliptic curves $y^{2}=x(x-2^{m}) (x+p)$
Tomasz Jędrzejak 1   ; Małgorzata Wieczorek 1
1 Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland
Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 53-67 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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We provide a description of the integral points on elliptic curves $y^{2}=x(x- 2^{m}) \times (x+p)$, where $p$ and $p+2^{m}$ are primes. In particular, we show that for $m=2$ such a curve has no nontorsion integral point, and for $m=1$ it has at most one such point (with $y \gt 0$). Our proofs rely upon numerical computations and a variety of results on quartic and other diophantine equations, combined with an elementary analysis.
DOI : 10.4064/ba8152-1-2019
Keywords: provide description integral points elliptic curves x times where primes particular curve has nontorsion integral point has point proofs rely numerical computations variety results quartic other diophantine equations combined elementary analysis

Tomasz Jędrzejak  1   ; Małgorzata Wieczorek  1

1 Institute of Mathematics University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland 
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Tomasz Jędrzejak; Małgorzata Wieczorek. Integral points on elliptic curves $y^{2}=x(x-2^{m}) (x+p)$. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 67 (2019) no. 1, pp. 53-67. doi: 10.4064/ba8152-1-2019

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