A COMPLETE CLASSIFICATION OF THE MERSENNE’S PRIMES AND ITS IMPLICATIONS FOR COMPUTING
Yeisson Alexis Acevedo Agudelo1
1Magister en Matemáticas Aplicadas, Universidad EAFIT, yaceved2@eafit.edu.co, ORCIDiD: https://orcid.org/0000-0002-1640-9084
ABSTRACT
A study of Mersenne’s primes is carried out using the multiplicative group modulo 360 and a complete classification is obtained by its residual classes. This allows the search for Mersenne’s primes to be classified into four subgroups mutually exclusive (disjoint) and contributes to the ordered selection of exponents to be computationally tested. According to this idea, Mersenne’s trapeze is presented with the purpose of giving a geometric representation for this classification. Finally, from one of the theorems presented and proven for primes in modulo 360, a conjecture is established that could be solved by computing.
Keywords: Mersenne’s prime; residual classes; Mersenne’s trapeze.
Recibido: 15 de Octubre de 2020. Aceptado: 7 de Diciembre de 2020
Received: October 15, 2020. Accepted: December 7, 2020
UNA CLASIFICACIÓN COMPLETA DE LOS NÚMEROS PRIMOS DE MERSENNE Y SUS IMPLICACIONES PARA LA COMPUTACIÓN
RESUMEN
Se realiza un estudio de los números primos de Mersenne utilizando el grupo multiplicativo módulo 360 y se obtiene una clasificación completa mediante sus clases residuales. Esto permite clasificar la búsqueda de los números primos de Mersenne en cuatro subgrupos mutuamente excluyentes (disjuntos) y contribuye a la selección ordenada de exponentes a probar computacionalmente. Acorde a esta idea, el trapecio de Mersenne se presenta con el propósito de dar una representación geométrica para esta clasificación. Finalmente, a partir de uno de los teoremas presentado y demostrado para primos en módulo 360, se establece una conjetura que podría resolverse mediante verificación computacional.
Palabras Clave: Primos de Merssene; Clases residuales; Trapecio de Mersenne.
Cómo citar este artículo: Y. Acevedo. “A complete classification of the Mersenne’s primes and its implications for computing”, Revista Politécnica, vol.16, no.32 pp.111-119, 2020. DOI:10.33571/rpolitec.v16n32a10
1. INTRODUCTION
A selection of problems in the theory of numbers focuses on mathematical problems within the boundaries of geometry and arithmetic, among these the Mersenne’s primes stand out, due to their computational search and incomprehensible randomness and the finding of even perfect number and it connection to perfect numbers and cryptography [1, 2, 3, 4].
A Mersenne’s prime is a prime number of the form
, where ρ is also a prime number. The current search
for Mersenne’s primes
, is led by the GIMPS computational project
(www.mersenne.org) and to date, only 51 of these numbers have been found. In
this work, a study for the Mersenne’s primes under the multiplicative group
modulo 360 is presented with the objective of contributing to this search by
improving the selection processes of the prime exponents and provide a complete
classification of these numbers. The methods used below are proper and basic
under a certain level of number theory [5].
2. CONTEXTUALIZATION
Let be a Mersenne’s prime.
Definition 1 (Ova-angular residue de ). Let be
and
prime numbers. The solution for
of the equation
and
, it will be called Ova-angular of
or
respectively and will be denoted by
,
, such that:
Notice that , is the residue that leaves a prime number when it is
divided by some integer, in this case by 360.
Definition 2 ( frequency rotation). Let be
the set of prime numbers. If
, then is said that its frequency of rotation denoted
, is given by the integer part of
when divided by 360.
From Definitions 1 y 2, it is true that
Definition 3 (Ova-angular function). Let be such that if
then
and
It is clear that
the function is well defined, in particular
is surjective.
2.1 Complete Classification
Theorem 1 ( Mersenne’s Primes). Let
be a prime number and
a
Mersenne’s prime. If
is the set formed by the Ova-angular
of the Mersenne primes
. Then:
Proof. Let be
a Mersenne’s prime, then there exists
, such that
. Next we analyze which of the 99 elements
are
possible to be Mersenne’s primes, prior to this the following criteria will be
taken into account:
Criterion A. Given the expression it
has to
, which
indicates for
that
must
be divisible by 2, 4 and 8.
Criterion B. Given the expression it
is possible in certain cases that
is a multiple of 3 then it has the expression
. Taking
common factor 3 we have that
, which is not possible since it contradicts the
fundamental theorem of arithmetic and also 3 does not divide any power of 2.
Consequently, this criterion affirms that
cannot
be a multiple of 3.
Criterion C. Given the expression it
is possible in certain cases that
is a
multiple of 5 then it has the expression
. Taking common factor 5 we have that
, which is not possible since
ends in 5 or 0, and the powers of 2 end in 2, 4, 8,
6, that is; according to the fundamental theorem of arithmetic, 5 does not
divide any power of 2. Consequently, this criterion states that
cannot
be a multiple of 5.
Criterion D. Let be,
so for these values in the expression
) it is conditioned that
must
end in the digit 4. Then
, this
is
, then
is even which would be contradictory since
is odd prime. Consequently, this criterion affirms
that
.
Criterion E. Let be, so for this value in equality
, we have infinite solutions to
with
. Which is contradictory to the fact that
must be prime. Consequently, this criterion affirms
that
.
Theorem 2 (Singular Mersenne). The number of Mersenne primes in class and in class
equals 1, i.e. the only Mersenne’s primes with
residues 3 and 7 are
and in
respectively.
Proof. For the of the Mersenne primes you get that
for
some
integer. Suppose that there is another prime
with
the same residue and
. Then this prime must satisfy for
integer:
a pure fractional.
Note. is a pure fractional since
ends in
, so it would never be a multiple of
.
Then is an integer and
is a pure fractional
. Then the only Mersenne’s prime in the class
is
.
Now, for the class of Mersenne primes, we have to:
. Suppose that there is another prime number
with the same residue and
. Then this prime must satisfy for
integer:
Since is an integer then
it must end in
or
, but since it never ends in
,
then the only option is that it must end in
, so
must end in
. Then
, then
is an even prime number
. Then the only Mersenne’s prime in the class
is
.
Figure 1 shows all the residuals of the prime numbers
greater than in modulo
and shows the isosceles trapeze that is formed by
joining Mersenne’s residues with an exponent greater than
. Hereafter this trapeze will be called Mersenne’s
trapeze.
Figure 1. Mersenne’s trapeze.
Theorem 3. Let be, then Mersenne’s primes in the class
are in the sequence:
where
and
Proof. Since then we have that
, now since
is a natural number, then it is convenient analyze
the values of
for which
intersects with some natural. Let it be
,
then it is clear that
,
sequence in which
we have the only integer solutions of the required logarithm.
Note.
It’s clear that if is prime then it is also a Mersenne’s prime and
according to Dirichlet’s theorem, if there are infinite values of
that
make prime numbers, then there are infinite Mersenne’s primes.
Corollary 2. For the class of Mersenne’s primes, the only
for the prime exponent
are:
.
Proof. Analogous to the previous Corollary (1)
proof, only that
Note.
It’s clear that if is prime then it is also a Mersenne’s prime, and
according to Dirichlet’s theorem, if there are infinite values of
that make prime numbers, then there are infinite
Mersenne’s primes.
Theorem 5. Let be, the Mersenne’s primes in the class
are in the sequence:
where
and
Proof. Similar to
the proof of the previous Theorem ().
Corollary 3. For the class of Mersenne’s primes, the only
for the prime exponent
are:
.
Proof. Analogous to the previous Corollary () proof, only that
.
Note. It’s clear that if is prime then it is also a Mersenne’s prime, and
according to Dirichlet’s theorem, if there are infinite values of
that make prime numbers, then there are infinite
Mersenne’s primes.
Theorem 6. Let be, then Mersenne’s primes in the class
are in the sequence:
where
and
Proof. Similar to
the proof of the previous Theorem ().
Corollary 4. For the class of Mersenne’s primes, the only
for the prime exponent
are:
.
Proof. Analogous to the previous Corollary () proof, only that
.
Note.
It’s clear that if is prime then it is also a Mersenne’s prime, and
according to Dirichlet’s theorem, if there are infinite values of
that make prime numbers, then there are infinite
Mersenne’s primes.
All established subgroups for each Mersenne’s prime family
are disjoint. This result is consistent with the one
presented in [6].
The reader is invited to articulate the previous
result in a computational way, with the Lucas-Lehmer test or use the Elliptic
curve test method presented in [7], to find a new Mersenne prime searching
and testing primes in each of the subgroups presented.
2.2 On residues modulo 360
Theorem 7 (Ova-Ova-Prime-Ova). Let be a prime number, let be
a completed set of residues of
mod
,
let be
y
;
arbitrary
prime numbers. Be also,
and
prime numbers such that
then, at least one of the following is true
Proof. (by reduction to absurdity). Let
be a prime number, let be
a completed set of residues of
mod
,
let be
y
;
arbitrary
prime numbers. Be also,
and
prime numbers such that
,
it is clear that , where
is a finite set. Suppose that none of the following
is true (Denial of thesis):
So, substituting in
and
after associating and canceling some terms, we have to
The expression , would be equivalent to affirming that for every
combination
of arbitrary prime numbers, there is no prime that is
the result of
minus another prime number, which is contradictory
with
the theorem proven by Harald Helfgott in [8], for example, there are the contradictions: Thus, it is true that since
are arbitrary prime numbers, in one or some cases the
primary arithmetic sum
will
also be a prime number.
Thus, we arrive at this contradiction for having
denied the thesis, then the assumption initial is false.
Example Theorem 7 (Ova-Ova-Prime-Ova)
Theorem establishes that at least one occurs, in this case as
it was chosen at random, seven numbers have turned out, it is fully fulfilled.
Conjecture 1 (The number of primes in the Ova-Ova-Prime-Ova
combination). Although the number of prime numbers resulting from theorem () is
at least one, at most, it should be 13.
The reader is invited to verify or solve the previous conjecture in a computational way.
3. CONCLUSIONS
A complete
classification of the Mersenne’s primes was presented through the
multiplicative group modulo . A
geometric representation of the residual classes generating these numbers was
obtained. It is possible to continue this work analyzing other applications
that this theory presents using the geometric properties for different prime
numbers, also is possible articulated through computational methods this theory
with different primality tests for these numbers finding a new Mersenne’s
primes (Lucas-Lehmer test or the Elliptic curve test method). The conjecture
about primes in this residual classes was established and the reader is invited
to verify or solve the proposed conjecture in a computational way.
4. ACKNOWLEDGEMENT
The author thanks the EAFIT university for all the support in his research.
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