Geodesic
Integral points on the elliptic curve $y^2=x^3-4p^2x$
Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 853-862

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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 $p\equiv -1\pmod 8$.
DOI : 10.21136/CMJ.2019.0529-17
Classification : 11D25, 11G05, 11Y50
Keywords: elliptic curve; integral point; quadratic equation; quartic Diophantine equation 
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     title = {Integral points on the elliptic curve $y^2=x^3-4p^2x$},
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Yang, Hai; Fu, Ruiqin. Integral points on the elliptic curve $y^2=x^3-4p^2x$. Czechoslovak Mathematical Journal, Tome 69 (2019) no. 3, pp. 853-862. doi: 10.21136/CMJ.2019.0529-17

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