MERSENNE NUMBERS TREASURE MAP

17 02 2009

Mersenne Numbers Treasure Map

The Mersenne generating function splits the integer set in some subsets:

\displaystyle \mathbb{N}\longrightarrow\mathbb{N}

\displaystyle f(n)\longrightarrow{2^{n}-1=M_n}

1-INTEGERS PARTITION SET2-RANGE PARTITION SET

\displaystyle f(0)=0

\displaystyle f(1)=1

\displaystyle f(Primes)\cap f(Composites) = \emptyset

\displaystyle f(Primes) \subset (Square Free \; \cup Composite factors of unknown Wieferich Primes )

\displaystyle f(Primes) \cap Primes = \textrm{Prime Mersenne Numbers}

\displaystyle f(Composites)\subset Composites

\displaystyle f(Composites)\cap Primes=\emptyset

\displaystyle f(Square Free)\subset (Composites \cup \{ 1 \} )

If \displaystyle n=a*b is a composite number, then \displaystyle M_{n}=M_{a*b} is also composite, because:
\displaystyle M_{a*b}=2^{a*b}-1=(2^{a}-1)\cdot (1+2^a+2^{2a}+\dots+2^{(b-1)a})=

\displaystyle M_{a}*\sum_{i=1}^b{2^{(b-i)\cdot a}}
And also if \displaystyle d|n , and if \displaystyle M_d , is not squarefree, then 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 \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]-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!

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