# Square of Wieferich prime is Poulet number

From Number

## 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.

## Particular cases

There are only two known Wieferich primes: 1093 and 3511. Hence, this fact gives only two Poulet numbers: and .

## 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 .