On the equation \(Y^2=X(X^2+p)\). (English) Zbl 0531.10014

If \(p\) is a prime \(\equiv 5\pmod 8\) E. Selmer’s conjecture [Math. Scand. 2, 49–54 (1954; Zbl 0055.27107)] predicts that the elliptic curve with Weierstrass equation \(y^2=x(x^2+p)\) has Mordell-Weil rank 1. In this paper the conjecture is verified for all such primes \(p<1000\). A table is provided. Some of the generators constructed are very large, e.g. for \(p=877\). The results also check with a conjecture of Mordell (which is equivalent to Selmer’s) on the quartic Diophantine equation \(r^4+ps^4=t^2\) [cf. L. J. Mordell, Q. J. Math., Oxf. II. Ser. 18, 1–6 (1967; Zbl 0154.29702)].
Reviewer: R.J.Stroeker


11G05 Elliptic curves over global fields
11D25 Cubic and quartic Diophantine equations
11Y50 Computer solution of Diophantine equations
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