Mersenne number for prime or Poulet implies prime or Poulet

From Number
Revision as of 00:17, 22 April 2009 by Vipul (Talk | contribs)

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

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:

.