# Definition:Strong Fibonacci Pseudoprime/Type I

## Definition

A strong Fibonacci pseudoprime of type I is a Carmichael number $N = \displaystyle \prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where:

 $\text {(1)}: \quad$ $\displaystyle 2 \paren {p_i + 1}$ $\divides$ $\displaystyle \paren {N - 1}$ for those $p_i$ of the form $4 m - 1$ $\text {(2)}: \quad$ $\displaystyle \paren {p_i + 1}$ $\divides$ $\displaystyle \paren {N \pm 1}$ for those $p_i$ of the form $4 m + 1$

where:

$N = \displaystyle \prod p_i$ is the prime decomposition of $N$
$\divides$ denotes divisibility.

## Source of Name

This entry was named for Leonardo Fibonacci.