The Mersenne generating function splits the integer set in some subsets:
$latex \displaystyle \mathbb{N}\longrightarrow\mathbb{N}$
$latex \displaystyle f(n)\longrightarrow{2^{n}-1=M_n} $
1-INTEGERS PARTITION SET
$latex \displaystyle Composites =\{4,6,8,9,10,12,...\}; \;[2]$
$latex \displaystyle Primes =\{2,3,5,7,11,13,...\}; \;[3]$
$latex \displaystyle Square Free =\{1,2,3,5,6,7,10,...\}; \;[4] $
$latex \displaystyle Integers =\{0,1,2,3,4,5,6,...\}=\mathbb{N}$
$latex \displaystyle \{0,1\} \cup Primes \cup Composites=Integers $
$latex \displaystyle Primes \subset Square Free$
$latex \displaystyle (Square Free-Primes-\{1\}) \subset Composites $
$latex \displaystyle Primes \cap Composites =\emptyset \\$
2-RANGE PARTITION SET
$latex \displaystyle f(0)=0$
$latex \displaystyle f(1)=1$
$latex \displaystyle f(Primes)\cap f(Composites) = \emptyset$
$latex \displaystyle f(Primes) \subset (Square Free \; \cup$ Composite factors of unknown Wieferich Primes $latex )$
$latex \displaystyle f(Primes) \cap Primes = \textrm{Prime Mersenne Numbers}$
$latex \displaystyle f(Composites)\subset Composites$
$latex \displaystyle f(Composites)\cap Primes=\emptyset$
$latex \displaystyle f(Square Free)\subset (Composites \cup \{ 1 \} )$
If $latex \displaystyle n=a*b$ is a composite number, then $latex \displaystyle M_{n}=M_{a*b}$ is also composite, because:
$latex \displaystyle M_{a*b}=2^{a*b}-1=(2^{a}-1)\cdot (1+2^a+2^{2a}+\dots+2^{(b-1)a})=$
$latex \displaystyle M_{a}*\sum_{i=1}^b{2^{(b-i)\cdot a}} $
And also if $latex \displaystyle d|n $ , and if $latex \displaystyle M_d$ , is not squarefree, then $latex M_n$, 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 $latex \displaystyle \frac{M_{n^2}}{M_n}=\sum_{i=1}^n{2^{(n-i)\cdot n}} $ on link [5]
Archives:
[a]-021709-MERSENNE NUMBERS TREASURE MAP.ppt
References:
[1]- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
A000225: 2^n - 1. (Sometimes called Mersenne numbers, although that name is usually reserved for A001348.)
[2]- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
A002808: The composite numbers: numbers n of the form x*y for x > 1 and y > 1.
[3]- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
A000040: The prime numbers.
[4]- N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences.
A005117: Squarefree numbers: numbers that are not divisible by a square greater than 1.
[5]- Leroy Quet Apr 19 2007, The On-Line Encyclopedia of Integer Sequences.
A128889: a(n) = (2^(n^2) -1) /(2^n -1).
[6]- Labos E., The On-Line Encyclopedia of Integer Sequences.
A049094: 2^n - 1 is divisible by a square >
[7]-Wieferich primes and Mersenne primes Miroslav Kures, Wieferich@Home - search for Wieferich prime.
[8]-Pacific J. Math. Volume 22, Number 3 (1967), 563-564 Henry G. Bray and Leroy J. Warren.
