On the equation \(y^2=x(x - 2^m)(x+q - 2^m)\)
Let \(E_{q,m}\) be the elliptic curve defined by \(y^2=x(x-2^m)(x+q-2^m)\). As a generalization of the twin prime conjecture, one may conjecture that, for a given \(m\), there exist infinitely many primes \(q\) such that \(q-2^m\) is also a prime. Thus, it is plausible that the family \[ \{E_{q,m}\,;\, q, q-2^m \text{ are both odd primes}\} \] consists of infinitely many members. The authors study the Mordell-Weil rank of \(E_{q,m}\), and give fairly complete description of the rank using the standard 2-descent method when both \(q\), \(q-2^{m}\) are primes. They subsequently study various other properties of elliptic curves in this family, including the rank of their quadratic twists by prime numbers.
scientific article; zbMATH DE number 5151590