Using Fibonacci factors to create Fibonacci pseudoprimes

TL;DR

This paper explores the construction of Fibonacci pseudoprimes using Fibonacci factors, aiming to understand their properties and potential connections to Baillie-PSW pseudoprimes, but finds no successful examples.

Contribution

It introduces a method to generate Fibonacci pseudoprimes from Fibonacci factors and examines their properties in relation to known pseudoprimes.

Findings

Constructed Fibonacci pseudoprimes from Fibonacci factors.

Attempted to find Baillie-PSW pseudoprimes among these, but was unsuccessful.

Provides insights into the structure of Fibonacci pseudoprimes.

Abstract

Carmichael showed for sufficiently large $L$, that $F_L$ has at least one prime divisor that is $\pm 1({\rm mod}\, L)$. For a given $F_L$, we will show that a product of distinct odd prime divisors with that congruence condition is a Fibonacci pseudoprime. Such pseudoprimes can be used in an attempt, here unsuccessful, to find an example of a Baillie-PSW pseudoprime, i.e.\ an odd Fibonacci pseudoprime that is congruent to $\pm 2({\rm mod}\, 5)$ and is also a base-2 pseudoprime.