Euler pseudoprimes for half of the bases

TL;DR

This paper characterizes odd numbers that are Euler pseudoprimes for exactly half of the admissible bases, showing they are precisely the special Carmichael numbers, thus providing a clear classification.

Contribution

It establishes a precise equivalence between Euler pseudoprimes for half of the bases and special Carmichael numbers, clarifying their relationship.

Findings

Odd numbers with this property are exactly the special Carmichael numbers.

Provides a characterization linking Euler pseudoprimes and Carmichael numbers.

Enhances understanding of pseudoprime distributions in number theory.

Abstract

We prove that an odd number is an Euler pseudoprime for exactly one half of the admissible bases if and only if it is a special Carmichael number.