### Rank computations for the congruent number elliptic curves.

Rogers, Nicholas F. (2000)

Experimental Mathematics

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Rogers, Nicholas F. (2000)

Experimental Mathematics

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Joseph H. Silvermann (1982)

Inventiones mathematicae

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Armand Brumer, Oisín McGuinness (1992)

Inventiones mathematicae

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Leopoldo Kulesz (2003)

Acta Arithmetica

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

Journal of Integer Sequences [electronic only]

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Kumiko Nakata (1979)

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Delaunay, C., Duquesne, S. (2003)

Experimental Mathematics

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Hizuru Yamagishi (1998)

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Bremner, Andrew (2000)

International Journal of Mathematics and Mathematical Sciences

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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.

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. ...

Koh-ichi Nagao (1997)

Manuscripta mathematica

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Mark Watkins, Stephen Donnelly, Noam D. Elkies, Tom Fisher, Andrew Granville, Nicholas F. Rogers (2014)

Publications mathématiques de Besançon

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We report on a large-scale project to investigate the ranks of elliptic curves in a quadratic twist family, focussing on the congruent number curve. Our methods to exclude candidate curves include 2-Selmer, 4-Selmer, and 8-Selmer tests, the use of the Guinand-Weil explicit formula, and even 3-descent in a couple of cases. We find that rank 6 quadratic twists are reasonably common (though still quite difficult to find), while rank 7 twists seem much more rare. We also describe our inability...

Sungkon Chang (2006)

Acta Arithmetica

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