Infinitude of Poulet numbers

Statement

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

$2^{n-1}\equiv 1(\mathrm{mod}n)$.

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 $n$ is a Poulet number, the Mersenne number $M_n$ 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 $a$ is a Poulet number. Consider the sequence:

$a_0=a,a_{i+1}=M_{a_i}$.

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 $M_{11}=2047=23\cdot 89$. This is not prime, so by fact (1), it is a Poulet number, and this completes the proof.