Catalan pseudoprime
# Catalan pseudoprime

In mathematics, a Catalan pseudoprime is an odd composite number n satisfying the congruence

    $(-1)^{\frac {n-1}{2}}\cdot C_{\frac {n-1}{2}}\equiv 2{\pmod {n}},$

where $C_m$ denotes the m-th Catalan number. The congruence also holds for every odd prime number n that justifies the name pseudoprimes for composite numbers n satisfying it.

## Properties

The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 (sequence A163209 in the OEIS) with the latter two being squares of Wieferich primes. In general, if p is a Wieferich prime, then $p^2$ is a Catalan pseudoprime.

