On the equation 𝑌²=𝑋(𝑋²+𝑝)

Andrew Bremner, J. W. S. Cassels

Type: Article

Publication Date: 1984-01-01

Citations: 246

DOI: https://doi.org/10.1090/s0025-5718-1984-0726003-4

Abstract

Generators are found for the group of rational points on the title curve for all primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p identical-to 5 left-parenthesis mod 8 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>5</mml:mn> <mml:mspace width="thickmathspace" /> <mml:mspace width="0.667em" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em" /> <mml:mn>8</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p \equiv 5\;\pmod 8</mml:annotation> </mml:semantics> </mml:math> </inline-formula> less than 1,000. The rank is always 1 in accordance with conjectures of Selmer and Mordell. Some of the generators are rather large.

Locations

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