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]

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Archives:

[a]-021709-MERSENNE NUMBERS TREASURE MAP.ppt

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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.

FERMAT AND MERSENNE NUMBERS CONJECTURE-(3)

(2)-CONSTRUCTING SOME FUNCTION ZEROS (ii)

(2.5) POWERS OF A PRIME * ODD SQUAREFREE:

$latex \displaystyle f(N)=f(p_{1}^{k}*M)=f(p_{1}^{k}*p_2*...*p_n)=0\;\rightarrow (k+1)|p_{1}^{k-1}* \phi\left(\frac{N}{p_{1}^{k-1}}\right) $, then the zeros can fall into two cases:

(2.5.1) Case: $latex \displaystyle (k+1)| \phi\left(\frac{N}{p_{1}^{k-1}}\right) $

Then if $latex \displaystyle d_i| \phi\left(\frac{N}{p_{1}^{k-1}}\right) $, $latex \displaystyle f(p_{1}^{d_{i}-1}*M)=0$, and we can built $latex \displaystyle \sigma_{0} \left(\frac{N}{p_{1}^{k-1}}\right) $, zeros,

one for every divisor of the product applied to every distinct prime factor

$latex \displaystyle \prod_{i=1}^{i=n}{ (p_{i}-1)}$ of $latex \displaystyle N$

Note that, in the particular case:

$latex \displaystyle (k+1)=(p_{i}-1) \rightarrow{k=p_{i}-2}\rightarrow{f(p_{1}^{p_{i}-2}*M)=0}$.

And $latex \displaystyle f(p_{1}^{p_{1}-1}*M)=Mod(p_{1}^{p_{1}-2}*\phi\left(\frac{N}{p_{1}^{k-1}}\right),p_{1})=0$.

(2.5.2) Case: $latex \displaystyle (k+1)|p_{1}^{k-1}$

$latex \displaystyle f(p_{1}^{p_{1}^{n}-1})=0$

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FERMAT AND MERSENNE NUMBERS CONJECTURE-(2)

(2)-CONSTRUCTING SOME FUNCTION ZEROS

$latex f(n)=Mod( \phi(n),\sigma_0(n))$

(2.1) PRIMES:

$latex \displaystyle f(p)=Mod(p-1,2)=0$, holds for every odd prime $latex p \in \mathbb{P} -\{2\}$.

$latex f(2)=1$

(2.2) PRODUCT OF DISTINCT PRIMES NOT 2:

$latex \displaystyle f(p_1*p_2*...*p_n)=0$, because $latex \displaystyle \sigma_0(p_i)|\phi(p_i) \rightarrow {2 |(p_i-1)}$, always holds if $latex \displaystyle p_{i} \neq 2$

With $latex \displaystyle k$, distinct primes, none of them equal two, it is possible to combine them in $latex \displaystyle 2^n$, products to find $latex \displaystyle 2^n$ zeros.

This set of zeros can be described as odd squarefree numbers [1].

(2.3) POWERS OF 2:

$latex \displaystyle f(2^k)=0\;\rightarrow (k+1)|2^{k-1}$, so $latex \displaystyle k+1=2^n$ must be a power of 2:

$latex \displaystyle k=2^n-1=M_n$
A power of 2, is a function zero, iff the exponent is a Mersenne number.

$latex \displaystyle f(2^{M_n})=f(2^{2^{n}-1})=0$

(2.4) POWERS OF A PRIME:

$latex \displaystyle f(p^k)=0\;\rightarrow (k+1)|(p-1)*p^{k-1}$, then the zeros can fall into two cases:

(2.4.1) Case: $latex \displaystyle (k+1)|(p-1)$

Then if $latex \displaystyle d|(p-1)$, $latex \displaystyle f(p^{d-1})=0$, and we can built $latex \displaystyle \sigma_{0}(p-1)$, zeros, one for every divisor of $latex \displaystyle (p-1)$.

Note that, in the particular case:

$latex \displaystyle (k+1)=(p-1) \rightarrow{k=p-2}\rightarrow{f(p^{p-2})=0}$.

And $latex \displaystyle f(p^{p-1})=Mod((p-1)*p^{p-2},p)=0$.

(2.4.2) Case: $latex \displaystyle (k+1)|p^{k-1}$

This is more general than (2.3):

$latex \displaystyle f(p^{p^{n}-1})=0$

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Archives:
[a]- 021409-FERMAT AND MERSENNE NUMBER CONJECTURE-(2).nb

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References:

[1]-CRCGreathouse at My Math Forum/Number Theory: Mersenne and Fermat Numbers congruence

FERMAT AND MERSENNE NUMBERS CONJECTURE-(1)

(1)-INITIAL IDEAS:

Playing, like always, with my computer, I´ve been plotting this function, that includes the Euler totient function, and the Divisor function.

$latex \displaystyle f(n)=Mod( \phi(n),\sigma_0(n))\: , \; n\in\mathbb{N_{*}^{+}}$

The first 25 values of $latex \displaystyle f(n)$, (not in OEIS) are:

$latex \displaystyle \{0, 1, 0, 2, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 3, 0, 0, 0, 2, 0, 2, 0, 0, 2,...\}$

Applying $latex \displaystyle f(x)$ to the Fermat Numbers, $latex \displaystyle F_n=2^{2^{n}}+1$, and to the Mersenne Numbers, $latex \displaystyle M_n=2^n-1$, we can conjecture the following congruences:

(1) $latex \displaystyle \phi(F_n) \equiv 0\; (mod \; \sigma_0(F_n))$

(2) $latex \displaystyle \phi(F_n-2) \equiv 0\; (mod \; \sigma_0(F_n-2))$

(3) $latex \displaystyle \phi(M_n) \equiv 0\; (mod \; \sigma_0(M_n))$

(4) $latex \displaystyle \phi(M_n+2) \equiv 0\; (mod \; \sigma_0(M_n+2))$

¿How do they can be proved? [1]

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Archives:

[a]-020809-FERMAT AND MERSENNE NUMBER CONJECTURE-(1).nb

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References:

[1]-My Math Forum/Number Theory: Mersenne and Fermat Numbers congruence