## Tuesday, 17 February 2009

### MERSENNE NUMBERS TREASURE MAP

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:

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.