On the equation 𝑌²=𝑋(𝑋²+𝑝) 论文

摘要

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.