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

Bremner, A.; Cassels, J. W. S.

Math. Comput. 42, 247-264 (1984).

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

MSC:

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

Keywords:

generators; rational points; Mordell-Weil rank one; Selmer conjecture; Mordell conjecture; elliptic curve; cubic Diophantine equation; quartic Diophantine equation

Citations:

Zbl 0055.27107; Zbl 0154.29702