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August 28th, 2018, 06:14 PM   #1
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Post 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.


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Last edited by Sebastian Garth; August 28th, 2018 at 06:51 PM. Reason: simplified
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August 29th, 2018, 04:18 PM   #2
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Quote:
Originally Posted by Sebastian Garth View Post
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.


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Today I learned.
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August 29th, 2018, 05:27 PM   #3
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Originally Posted by Maschke View Post
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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