Girish Mahajan (Editor)

Catalan pseudoprime

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

( − 1 ) n − 1 2 ⋅ C n − 1 2 ≡ 2 ( mod n ) ,

where Cm 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.

References

Catalan pseudoprime Wikipedia
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