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On the equation $$Y^ 2=(X+p)(X^ 2+p^ 2)$$. (English) Zbl 0810.11038
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.

##### MSC:
 11G05 Elliptic curves over global fields 11D25 Cubic and quartic Diophantine equations
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##### References:
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