Infinitude of Poulet numbers

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Template:Infinitude fact

Statement

There exist infinitely many Poulet numbers, i.e., there infinitely many odd composite numbers such that:

.

Facts used

  1. Mersenne number for prime or Poulet implies prime or Poulet
  2. Mersenne number for composite number is composite

Proof

By fact (1), if is a Poulet number, the Mersenne number is either prime or a Poulet number. But by fact (2), it cannot be prime. Thus, the Mersenne number for a Poulet number is a Poulet number.

Thus, suppose is a Poulet number. Consider the sequence:

.

In other words, each member of the sequence is the Mersenne number for the preceding number. This is a strictly increasing sequence, and each member of it is a Poulet number, so if there is one Poulet number, there are infinitely many Poulet numbers.

Thus, it is sufficient to find just one Poulet number. Using fact (1), consider . This is not prime, so by fact (1), it is a Poulet number, and this completes the proof.