Catalan pseudoprime

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 above congruence holds for every odd prime number n, so any composite n that satisfies it is pseudoprime.

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.

References

• 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.