Catalan pseudoprime

This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Catalan pseudoprime" – news · newspapers · books · scholar · JSTOR (December 2015) (Learn how and when to remove this message)

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 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.

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.

• 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.