## 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.

### MSC:

 11G05 Elliptic curves over global fields 11D25 Cubic and quartic Diophantine equations 11Y50 Computer solution of Diophantine equations
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### References:

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