# Definition:Strong Fibonacci Pseudoprime

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## Definition

A **strong Fibonacci pseudoprime** is a Carmichael number which also satisfies one of the following conditions:

### Type I

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 $4 m + 1$ |

### Type II

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

- $2 \paren {p_i + 1} \divides \paren {N - p_i}$ for all $p_i$

## Also known as

Some sources refer to such a number as a **strong $\left({-1}\right)$-Dickson pseudoprime**, for Leonard Eugene Dickson.

## Also see

- Results about
**strong Fibonacci pseudoprimes**can be found here.

## Source of Name

This entry was named for Leonardo Fibonacci.

## Sources

- Jul. 1993: R.G.E. Pinch:
*The Carmichael Numbers up to $10^{15}$*(*Math. Comp.***Vol. 61**,*no. 203*: pp. 381 – 391) www.jstor.org/stable/2152963