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2020-10-17, 23:52   #12
a1call

"Rashid Naimi"
Oct 2015
Remote to Here/There

2×991 Posts

Quote:
 Originally Posted by Viliam Furik No, all those Mersenne numbers you listed have one factor and a (definitely) composite cofactor, so at least three factors.
I see, thanks for the explanation.

"PRP Cofactor" title is probably not the best choice of words then.

2020-10-18, 03:08   #13
mathwiz

Mar 2019

11·13 Posts

Quote:
 Originally Posted by a1call If I read the merssene.org correctly, M117239 has no known factors and F117239 is a (Probable) semiprime. So any complete list will have to be exclusively less than 117239.
A small notational quibble: doesn't $F_n$ typically denote the n-th Fermat number, i.e. $2^{2^n}+1$?

 2020-10-18, 03:22 #14 a1call     "Rashid Naimi" Oct 2015 Remote to Here/There 2·991 Posts I'm sure it does. I never hid the fact that I am no mathematician. Thank you for the correction. I will try to remember the notation. Last fiddled with by a1call on 2020-10-18 at 03:23
2020-10-18, 09:23   #15
Viliam Furik

"Viliam FurÃ­k"
Jul 2018
Martin, Slovakia

5·67 Posts

Quote:
 Originally Posted by a1call I see, thanks for the explanation. "PRP Cofactor" title is probably not the best choice of words then.
It is, because if cofactor (the rest of the number, when divided by known certainly-prime factors) is PRP, then if the number has only one certainly-prime factor, it means it is most probably a semiprime.

If the cofactor is tested composite by PRP test, the cofactor is certainly composite.

If you refer to this page, then yes, it may be kind of confusing to see PRP cofactor written there, but that's really only a title. If it would be PRP, for real, it would have had written "PRP_PRP_PRP_PRP_" in place of residue.

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