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Keywords:
elliptic curve; integral point; quadratic equation; quartic Diophantine equation
elliptic curve; integral point; quadratic equation; quartic Diophantine equation
Summary:
Let $p$ be a fixed odd prime. We combine some properties of quadratic and quartic Diophantine equations with elementary number theory methods to determine all integral points on the elliptic curve $E\colon y^2=x^3-4p^2x$. Further, let $N(p)$ denote the number of pairs of integral points $(x,\pm y)$ on $E$ with $y>0$. We prove that if $p\geq 17$, then $N(p)\leq 4$ or $1$ depending on whether $p\equiv 1\pmod 8$ or $p\equiv -1\pmod 8$.
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