Square of Wieferich prime is Poulet number

Statement

Suppose $\text{[math]}$ is a Wieferich prime, i.e., a prime number such that:

$\text{[math]}$

Then, $\text{[math]}$ is a Poulet number (also called Sarrus number), i.e., a Fermat pseudoprime to base 2.

There are only two known Wieferich primes: 1093 and 3511. Hence, this fact gives only two Poulet numbers: $\text{[math]}$ and $\text{[math]}$ .

Given: $\text{[math]}$ is a prime such that $\text{[math]}$

To prove: $\text{[math]}$

Proof: We have:

$\text{[math]}$

Thus, $\text{[math]}$ divides $\text{[math]}$ . This gives that:

$\text{[math]}$

Combining this with the given information, we get that $\text{[math]}$ .