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

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.

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+1 for the wordplay. –  Willie Wong Jan 17 '11 at 17:35
It introduced a typo and made it illegible without mathjax. Mathjax is quite jittery and is completely overkill when html suffices. –  Jack Schmidt Jan 17 '11 at 17:36
Its not German sofa its Sophie Germain –  anonymous Jan 17 '11 at 17:49
Sophie Germain was much better with primes than me. My numbers are more like a sectional sofa. One section with 2, one with (q+1)/2, and one with (qq+q+1)/2. –  Jack Schmidt Jan 17 '11 at 18:07
I think you mean $(qq-q+1)/2$? Also, why are you factoring out $2$? $qq\pm q+1$ is odd. –  joriki Sep 13 '12 at 9:26

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