# Square of Wieferich prime is Poulet number

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

## 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 .