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 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 {{OEIS|A163209}} with the latter two being squares of Wieferich primes. In general, if p is a Wieferich prime, then p² is a Catalan pseudoprime.

References

{{cite journal | last1 = Aebi | first1 = Christian | last2 = Cairns |first2 = Grant | title = Catalan numbers, primes and twin primes

| journal = Elemente der Mathematik |volume=63 |issue=4 | pages = 153–164 | year = 2008 | url = http://gradelle.educanet2.ch/christian.aebi/.ws_gen/9/catalan.pdf | doi = 10.4171/EM/103| doi-access = free }}

[http://compmath.wordpress.com/catalan-pseudoprimes/ Catalan pseudoprimes]. Research in Scientific Computing in Undergraduate Education.

Category:Pseudoprimes