My Math Forum A new definition for a superset of Carmichael numbers

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 August 28th, 2018, 05:14 PM #1 Member   Joined: Jul 2010 Posts: 83 Thanks: 2 A new definition for a superset of Carmichael numbers We define a so-called Extended Korselt Pseudoprime in the following way: A composite number N is an Extended Korselt Pseudoprime iff for all prime divisors F of (N-1) the congruence F^(N-1) = 1 mod N holds true. Incidentally the theorem does have at least one practical application: the Lucas Primality test. Owing to the fact that the Lucas method already requires all of the factors of (N-1) to be known a priori we can therefore apply the above criterion to efficiently exclude all but the most "resilient" pseudoprimes before moving on to the classical Lucas primality test. ... Last edited by Sebastian Garth; August 28th, 2018 at 05:51 PM. Reason: simplified
August 29th, 2018, 03:18 PM   #2
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Quote:
 Originally Posted by Sebastian Garth We define a so-called Extended Korselt Pseudoprime in the following way: A composite number N is an Extended Korselt Pseudoprime iff for all prime divisors F of (N-1) the congruence F^(N-1) = 1 mod N holds true. Incidentally the theorem does have at least one practical application: the Lucas Primality test. Owing to the fact that the Lucas method already requires all of the factors of (N-1) to be known a priori we can therefore apply the above criterion to efficiently exclude all but the most "resilient" pseudoprimes before moving on to the classical Lucas primality test. ...

Today I learned.

August 29th, 2018, 04:27 PM   #3
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Quote:
 Originally Posted by Maschke Today I learned.
Very funny, smart ass.

Anyway I think I was a little rash in posting this (I'm getting old and just tend to think out loud these days). In retrospect I can't see any real justification for associating this with Korselt's theorem per se. Still pretty useful as a precursor to the Lucas primality test though.

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