Catalan pseudoprime explained

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

( -1 ) ^{(n -1)/(2)} ⋅ C _{(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 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 . Catalan numbers, primes and twin primes . Elemente der Mathematik . 63 . 4 . 153–164 . 2008 . 10.4171/EM/103. free .
• Catalan pseudoprimes. Research in Scientific Computing in Undergraduate Education.