On the fibres of an elliptic surface where the rank does not jump

For a non-constant elliptic surface over ${\mathbb{P}}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell–Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infinitely many Mersenne primes.