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 $p\equiv -1\pmod 8$.
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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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

[1] Bennett, M. A.: Integral points on congruent number curves. Int. J. Number Theory 9 (2013), 1619-1640. | DOI | MR | JFM

[2] Bennett, M. A., Walsh, G.: The Diophantine equation $b^2X^4-dY^2=1$. Proc. Am. Math. Soc. 127 (1999), 3481-3491. | DOI | MR | JFM

[3] Bremner, A., Silverman, J. H., Tzanakis, N.: Integral points in arithmetic progression on $y^2=x(x^2-n^2)$. J. Number Theory 80 (2000), 187-208. | DOI | MR | JFM

[4] Draziotis, K. A.: Integer points on the curve $Y^2=X^3\pm p^kX$. Math. Comput. 75 (2006), 1493-1505. | DOI | MR | JFM

[5] Draziotis, K., Poulakis, D.: Practical solution of the Diophantine equation $y^2= x\*(x+2^ap^b)\*(x-2^ap^b)$. Math. Comput. 75 (2006), 1585-1593. | DOI | MR | JFM

[6] Draziotis, K., Poulakis, D.: Solving the Diophantine equation $y^2= x(x^2-n^2)$. J. Number Theory 129 (2009), 102-121 corrigendum 129 2009 739-740. | DOI | MR | JFM

[7] Fujita, Y., Terai, N.: Integer points and independent points on the elliptic curve $y^2=x^3-p^kx$. Tokyo J. Math. 34 (2011), 367-381. | DOI | MR | JFM

[8] Fujita, Y., Terai, N.: Generators and integer points on the elliptic curve $y^2=x^3-nx$. Acta Arith. 160 (2013), 333-348. | DOI | MR | JFM

[9] Hardy, G. H., Wright, E. M.: An Introduction to the Theory of Numbers. The Clarendon Press, Oxford University Press, New York (1979). | MR | JFM

[10] Spearman, B. K.: Elliptic curves $y^2=x^3-px$ of rank two. Math. J. Okayama Univ. 49 (2007), 183-184. | MR | JFM

[11] Spearman, B. K.: On the group structure of elliptic curves $y^2=x^3-2px$. Int. J. Algebra 1 (2007), 247-250. | DOI | MR | JFM

[12] Tunnell, J. B.: A classical Diophantine problem and modular forms of weight $3/2$. Invent. Math. 72 (1983), 323-334. | DOI | MR | JFM

[13] Walsh, P. G.: Maximal ranks and integer points on a family of elliptic curves. Glas. Mat., III. Ser. 44 (2009), 83-87. | DOI | MR | JFM

[14] Walsh, P. G.: On the number of large integer points on elliptic curves. Acta Arith. 138 (2009), 317-327. | DOI | MR | JFM

[15] Walsh, P. G.: Maximal ranks and integer points on a family of elliptic curves II. Rocky Mt. J. Math. 41 (2011), 311-317. | DOI | MR | JFM

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