are 168434 and 21246794, 16843 and 2124679 are Wolstenholme primes . Besides, with an exception of 168432 and 21246792, no composites are known to hold for () with k = 2, much less k = 3. Thus the conjecture is considered likely because Wolstenholme's congruence seems over-constrained and artificial for composite numbers. Moreover, if the congruence does hold for any particular n other than a prime or prime power, and any particular k, it does not imply that The number of Wolstenholme pseudoprimes up to is , so the sum of reciprocals of those numbers converges. The constant follows from the existence of only three Wolstenholme pseudoprimes up to . The number of Wolstenholme pseudoprimes up to should be at least 7 if the sum of its reciprocals diverged, and since this is not satisfied, the counting function of these pseudoprimes is at most for some efficiently computable constant ; we can take as 499712. Generalizations Leudesdorf has proved that for a positive integer n coprime to 6, the following congruence holds: See also Fermat's little theorem Wilson's theorem Wieferich prime Wilson prime Wall–Sun–Sun prime List of special classes of prime numbers Table of congruences Notes References . . . . . . R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862—2012). . External links The Prime Glossary: Wolstenholme prime Status of the search for Wolstenholme primes Classes of prime numbers Factorial and binomial topics Articles containing proofs Theorems about prime numbers