# When is $\frac{a^p+1}{a+1}$ a pseudoprime to base $b$?

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

• $$a=5^3$$ , $$p=7$$

• $$a=13^2$$ , $$p=17$$