Catalan pseudoprime

Template:Short description Template:Refimprove In mathematics, a Catalan pseudoprime is an odd composite number n satisfying the congruence

$(-1{)}^{\frac{n-1}{2}}\cdot C_{\frac{n-1}{2}}\equiv 2(\mathrm{mod}n),$

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 Template: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

• Template:Cite journal
• Catalan pseudoprimes. Research in Scientific Computing in Undergraduate Education.