# Catalan pseudoprime

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

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

where Cm 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 p2 is a Catalan pseudoprime.