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Inmathematics, aCatalan pseudoprime is an oddcomposite numbern satisfying the congruence

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

whereC_m denotes them-thCatalan number.

The above congruence holds for every oddprime numbern, so any compositen that satisfies it ispseudoprime.

The only known Catalan pseudoprimes are: 5907, 1194649, and 12327121 (sequenceA163209 in theOEIS) with the latter two being squares ofWieferich primes. In general, ifp is a Wieferich prime, thenp² is a Catalan pseudoprime.

• Aebi, Christian; Cairns, Grant (2008)."Catalan numbers, primes and twin primes"(PDF).Elemente der Mathematik.63 (4):153–164.doi:10.4171/EM/103.
• Catalan pseudoprimes. Research in Scientific Computing in Undergraduate Education.

• Pseudoprimes

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