Let $a$ and $b$ be postive integers greater than $1$ and $p$ be an odd prime.

Is there an easy criterion whether the number $$N:=\frac{a^p+1}{a+1}$$ is a weak Fermat-pseudoprime to base $b$, in other words, whether $$b^{N-1}\equiv 1\mod N$$ holds ?

I am particular interested in the case $a=b^k$ with positive integer $k$. Even in this case, the congruence does not necessarily hold, examples are

  • $a=5^3$ , $p=7$

  • $a=13^2$ , $p=17$


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