Definition:Strong Fibonacci Pseudoprime

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 $N = \ds \prod p_i$ such that an even number of the prime factors $p_i$ are of the form $4 m - 1$ where:

\(\text {(1)}: \quad\)

\(\ds 2 \paren {p_i + 1}\)

\(\divides\)

\(\ds \paren {N - 1}\)

for those $p_i$ of the form $4 m - 1$

\(\text {(2)}: \quad\)

\(\ds \paren {p_i + 1}\)

\(\divides\)

\(\ds \paren {N \pm 1}\)

for those $p_i$ of the form $4 m + 1$

Type II

A strong Fibonacci pseudoprime of type II is a Carmichael number $N = \ds \prod p_i$ 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