### A search for high rank congruent number elliptic curves.

Dujella, Andrej, Janfada, Ali S., Salami, Sajad (2009)

Journal of Integer Sequences [electronic only]

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Dujella, Andrej, Janfada, Ali S., Salami, Sajad (2009)

Journal of Integer Sequences [electronic only]

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Rogers, Nicholas F. (2000)

Experimental Mathematics

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Armand Brumer, Oisín McGuinness (1992)

Inventiones mathematicae

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Julián Aguirre, Fernando Castañeda, Juan Carlos Peral (2000)

Revista Matemática Complutense

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Seven elliptic curves of the form y = x + B x and having rank at least 8 are presented. To find them we use the double descent method of Tate. In particular we prove that the curve with B = 14752493461692 has rank exactly 8.

Tesuji Shioda (1991)

Inventiones mathematicae

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J. E. Cremona (1993)

Journal de théorie des nombres de Bordeaux

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In this note we extend the computations described in [4] by computing the analytic order of the Tate-Shafarevich group III for all the curves in each isogeny class ; in [4] we considered the strong Weil curve only. While no new methods are involved here, the results have some interesting features suggesting ways in which strong Weil curves may be distinguished from other curves in their isogeny class.

Bremner, Andrew (2000)

International Journal of Mathematics and Mathematical Sciences

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Leopoldo Kulesz (2003)

Acta Arithmetica

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Hizuru Yamagishi (1998)

Manuscripta mathematica

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Joseph H. Silvermann (1982)

Inventiones mathematicae

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Kumiko Nakata (1979)

Manuscripta mathematica

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Sungkon Chang (2006)

Acta Arithmetica

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Farzali Izadi, Foad Khoshnam, Arman Shamsi Zargar (2016)

Colloquium Mathematicae

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We construct a family of elliptic curves with six parameters, arising from a system of Diophantine equations, whose rank is at least five. To do so, we use the Brahmagupta formula for the area of cyclic quadrilaterals (p³,q³,r³,s³) not necessarily representing genuine geometric objects. It turns out that, as parameters of the curves, the integers p,q,r,s along with the extra integers u,v satisfy u⁶+v⁶+p⁶+q⁶ = 2(r⁶+s⁶), uv = pq, which, by previous work, has infinitely many integer solutions. ...