The Mersenne generating function splits the integer set in some subsets:
1-INTEGERS PARTITION SET2-RANGE PARTITION SET
Composite factors of unknown Wieferich Primes
If is a composite number, then is also composite, because:
And also if , and if , is not squarefree, then , can not be squarefree [8].
The only known Wieferich primes are 1093 and 3511, but they can not be prime factors of a Mersenne prime, see [6] and [7].
Note: See on link [5]
Archives:
[a]-021709-MERSENNE NUMBERS TREASURE MAP.ppt
References:[1]-A000225 The On-Line Encyclopedia of Integer Sequences!
[2]-A002808 The On-Line Encyclopedia of Integer Sequences!
[3]-A002808 The On-Line Encyclopedia of Integer Sequences!
[4]-A005117 The On-Line Encyclopedia of Integer Sequences!
[5]-A128889 The On-Line Encyclopedia of Integer Sequences!
[6]-Wieferich primes and Mersenne primes Miroslav Kures, Wieferich@Home – search for Wieferich prime.
[7]-Pacific J. Math. Volume 22, Number 3 (1967), 563-564 Henry G. Bray and Leroy J. Warren.
[8]-A049094 The On-Line Encyclopedia of Integer Sequences!