Square of Wieferich prime is Poulet number

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Statement

Suppose is a Wieferich prime, i.e., a prime number such that:

Then, is a Poulet number (also called Sarrus number), i.e., a Fermat pseudoprime to base 2.

Proof

Given: is a prime such that

To prove:

Proof: We have:

Thus, divides . This gives that:

Combining this with the given information, we get that .