Prove that for any prime $q > 3$, $(q^q - (-1)^{(q-1)/2})/(q-(-1)^{(q-1)/2})$ is never prime (or disprove by counterexample). If this is true, is there a possible trivial factorization for $(q^q - (-1)^{(q-1)/2})/(q-(-1)^{(q-1)/2})$?
The first few examples appear to be
$q=5$ and $(5^5-1)/4=11*71$, $q=7$ and $(7^7+1)/8=113*911$, $q=11$ and $(11^{11}+1)/12=23*89*199*58367$, $q=13$ and $(13^{13}-1)/(13-1)=53*264031*1803647$.
Let $p*p_2$ be the non-trivial factorization of $(q^q - (-1)^{(q-1)/2})/(q-(-1)^{(q-1)/2})$. Then, for all odd primes $q$,
$q=3$, $p=1$, $p_2=7$
$q=5$, $p=11$, $p_2=71$
$q=7$, $p=113$, $p_2=911$
$q=11$, $p=58367$, $p_2=407353$
$q=13$, $p=1803647$, $p_2=13993643$
$p$ and $p_2$ seem to be of relative same size, and with $p_2$ being approximately $7.76p$.
Can anyone find the interesting pattern here?