On the equation \(Y^ 2=(X+p)(X^ 2+p^ 2)\) (Q1344011)

The family of elliptic curves over \(\mathbb{Q}\) given by the title equation is studied. It is shown that for \(p\) a prime number \(\equiv\pm 3\bmod 8\), the only rational solution to the equation given here is the one with \(y=0\). The same is true for \(p=2\). Standard conjectures predict that the rank of the group of rational points is odd for all other primes \(p\). A lot of numerical evidence in support of this is given. We show that the rank is bounded by 3 in general for prime numbers \(p\). Moreover, this bound can only be attained for certain special prime numbers \(p\equiv 1\bmod 16\). Examples of such rank 3 curves are given. Lastly for certain primes \(p\equiv 9\bmod 16\) non-trivial elements in the Shafarevich group of the elliptic curve are constructed. In the literature one finds similar investigations of elliptic curves depending on a prime number \(p\); however, usually these are elliptic curves with complex multiplication. It may be interesting to note that the curves considered here do not admit complex multiplication.