I noticed that Hardy and Wright in their "An Introduction to Theory of Numbers"(sixth edition) have asked the following:
Is it ever true that $2^{p-1}\equiv 1 \bmod p^2 \tag{*}\;\;\;?$
They have pointed out that for $p=1093$ there is a solution to $(*)$ .But they have stated that such $p$ are sparse .
Question: Do there exist infinitely many primes $p$ such that $a^{p-1}\equiv 1$ $\text{mod } p^2$? For some fixed $a\in Z^\mathbb{+}$ for $a>2$? Sorry if my question is absolutely trivial.