Mersenne number for prime or Poulet implies prime or Poulet
From Number
Statement
Suppose is a natural number such that:
.
Consider the Mersenne number
. Then, we have:
.
In other words, the Mersenne number for a number that is either a prime number or a Poulet number (i.e., an odd composite number that is pseudoprime to base ), is also either a prime number or a Poulet number.
Related facts
Applications
Proof
Given: A natural number such that
.
To prove: .
Proof: By assumption, divides
, so there exists an integer
such that
. Thus,
. Thus, we have:
.
Since divides
, we get that:
.