Integral points on the elliptic curve \(y^2=x^3-4p^2x\). (English) Zbl 1513.11137

In this paper, the integral points on the elliptic curve \(E\) defined by the equation \(y^2=x^3-4p^2x\), where \(p\) is a prime \(\geq 17\), are determined. The proof is divided in four cases according to the form of \(p\). For instance, if \(p=a^4+b^4\), then \((x,\pm y)=(-4a^2b^2,\pm 4ab|a^4-b^4|)\). The proofs are relied on some properties of quadratic and quartic Diophantine equations. If \(N(p)\) denotes the number of pairs of nontrivial integer points of \(E\) and \(p\equiv\pm 1\ (\bmod\ 8)\), then, it is proved that \(N(p)\leq 4\), if \(p\equiv 1\ (\bmod\ 8)\) and \(N(p)\leq 1\), if \(p\equiv -1\ (\bmod\ 8)\). Note that in case where \(p=17\), the nontrivial integral points of \(E\) are: \((x,\pm y)=(-16,\pm 120),(-2,\pm 48),(162,\pm 2016),(578,\pm 13872)\), and so, the upper bound for \(N(p)\) given above is attainable.


11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11Y50 Computer solution of Diophantine equations
Full Text: DOI


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