Wolstenholme and Morley, Primes and Pseudoprimes

TL;DR

This paper establishes equivalences between certain prime and pseudoprime properties related to Wolstenholme and Morley primes, highlighting the rarity and specific conditions of these special pseudoprimes.

Contribution

It proves the equivalence of Wolstenholme and Morley pseudoprimes of order 2 and characterizes the rarity of such pseudoprimes of order 1.

Findings

p^2 is a Morley pseudoprime iff p^2 is a Wolstenholme pseudoprime iff p is a Wolstenholme prime iff p is a Morley prime

Only 3 pseudoprimes of Wolstenholme's type of order 1 are known, none of Morley's type

The paper clarifies the relationship between Wolstenholme and Morley primes and pseudoprimes.

Abstract

In this note, we prove $p^2$ is a Morley pseudoprime (of order 2) iff $p^2$ is a Wolstenholme pseudoprime (of order 2) iff $p$ is a Wolstenholme prime iff $p$ is a Morley prime. Concerning pseudoprimes of order 1 that are not powers of primes, only 3 are known of Wolstenholme's type and absolutely none have yet been identified of Morley's type.