German sofa primes: Can both $q$ and $\frac{q^3+1}{2}$ be prime?

prime-numbers elementary-number-theory

Is there an odd prime integer $\displaystyle q$ such that $\displaystyle p= \frac{q^3+1}{2}$ is also prime?

A quick search did not find any, nor a pattern in the prime factorization of p. This is a possible quick solution to the unitary and Ree cases of ME.16954.


Isn't this divisble by $\displaystyle \frac{q+1}{2}$?

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