Catalan pseudoprime

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

(-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.

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.

gollark: Idea: what if you make a machine which will automatically open the box if an XK-class scenario occurs/is imminent?]

gollark: Probability fuses or something.

gollark: Although you could have a cool name for it.

gollark: basically the box thing but the same !!!!

gollark: http://www.scp-wiki.net/scp-1968

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.

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