T-Fermat integers

Abstract

We introduce the concept of a T-Fermat integer, which generalizes the notion of a prime number. We show that any composite T-Fermat integer, if one exists, must be a Carmichael number. We prove several properties of T-Fermat integers and conjecture that there are infinitely many composite T-Fermat integers. Together with a further structural conjecture, this suggests a possible route toward proving the infinitude of primes $p$ such that $\ddot{I}‰(p-1)\le 2$.

Revised version. Title changed because the term 'almost primes' is already established in literature. Presentation and readability improved throughout