Carmichael numbers and the sieve

TL;DR

This paper proves there are infinitely many Carmichael numbers with prime factors of a specific quadratic form using sieve methods.

Contribution

It establishes the infinitude of Carmichael numbers with prime factors of the form 1 + a^2 + b^2, a novel result in number theory.

Findings

Infinitely many Carmichael numbers exist with prime factors of the form 1 + a^2 + b^2.

Utilizes sieve techniques to prove the infinitude.

Connects Carmichael numbers with primes of a specific quadratic form.

Abstract

Using the sieve, we show that there are infinitely many Carmichael numbers whose prime factors all have the form $p = 1 + a^2 + b^2$ with $a,b \in{\mathbb Z}$.