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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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    Its not German sofa its **Sophie Germain**2011-01-17
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    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)/2.2011-01-17
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    I think you mean $(qq-q+1)/2$? Also, why are you factoring out $2$? $qq\pm q+1$ is odd.2012-09-13

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