Triads of integers with equal sums of squares and equal products and a related multigrade chain

Choudhry, Ajai; Wróblewski, Jarosław

doi:10.4064/aa8574-9-2016

Cited by 1 publication

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“…then the numbers a i , b i , c i , i = 1, 2, 3, 4, satisfy the diophantine chains (20). Several parametric solutions of the simultaneous diophantine equations (24) are given in [2]. These solutions yield parametrized families of the cubic model of elliptic curves with 11 known rational points.…”

Section: 2mentioning

confidence: 99%

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Symmetric diophantine systems and families of elliptic curves of high rank

Choudhry¹

2019

Rocky Mountain J. Math.

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While there has been considerable interest in the problem of finding elliptic curves of high rank over Q, very few parametrized families of elliptic curves of generic rank ≥ 8 have been published. In this paper we use solutions of certain symmetric diophantine systems to construct several parametrized families of elliptic curves with their generic ranks ranging from at least 8 to at least 12. Specific numerical values of the parameters yield elliptic curves with quite large coefficients and we could therefore determine the precise rank only in a few cases where the rank of the elliptic curve ≤ 13. It is, however, expected that the parametrized families of elliptic curves obtained in this paper would yield examples of elliptic curves with much higher rank.Mathematics Subject Classification: 11G05, 11D25, 11D41

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Section: 2mentioning

confidence: 99%

“…It has been shown in [2] that infinitely many three-parameter solutions of the diophantine chains (24) can be obtained. Each of these solutions yields a three-parameter solution of the diophantine chains (20), and hence we can obtain infinitely many families of elliptic curves whose coefficients are given in terms of three arbitrary parameters.…”

Section: 2mentioning

confidence: 99%