Renaud and Henri Lifchitz

mentioned this relation in a 2000 paper, see section 4. Note: they use N

_{p} to denote what we call W

_{p}.

As they point out, this relation means that if W

_{2n+1} is PRP, then if either W

_{n} or M

_{n} are fully factored, then W

_{2n+1} can be proven prime by the N−1 method. (Note n does not need to be prime, only 2n+1).

For instance, if we could fully factor M

_{47684} or W

_{47684} then we could prove that W

_{95369} is not just PRP but prime. Spoiler alert: they are nowhere near fully-factored.

Or we could look at all Mersenne primes M

_{p} for which 2p+1 is also prime (

OEIS A065406) and check to see if any unfactored W

_{2p+1} test PRP. Spoiler alert: they don't. M

_{43,112,609} is a Mersenne prime but W

_{86,225,219} is composite.

In short, this relation doesn't have much practical use.