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Old 2019-11-23, 00:51   #2
GP2
 
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Sep 2003

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Renaud and Henri Lifchitz mentioned this relation in a 2000 paper, see section 4. Note: they use Np to denote what we call Wp.

As they point out, this relation means that if W2n+1 is PRP, then if either Wn or Mn are fully factored, then W2n+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 M47684 or W47684 then we could prove that W95369 is not just PRP but prime. Spoiler alert: they are nowhere near fully-factored.

Or we could look at all Mersenne primes Mp for which 2p+1 is also prime (OEIS A065406) and check to see if any unfactored W2p+1 test PRP. Spoiler alert: they don't. M43,112,609 is a Mersenne prime but W86,225,219 is composite.

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

Last fiddled with by GP2 on 2019-11-23 at 01:01
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