Suppose that every odd $n$ is the sum of a squarefree number and a power of 2. Then the set of primes
$p$ such that $2 ^ p ≡ 2 \mod p ^ 2$ is infinite. This is Theorem 1 in [GrSo98].
[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.
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