A COMPLETE CLASSIFICATION OF THE MERSENNE’S PRIMES

AND ITS IMPLICATIONS FOR COMPUTING

Yeisson Alexis Acevedo Agudelo^{1}

^{ }

^{1}Magister
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.

5. REFERENCES

[1] I. N. S. Waclaw Sierpinski, M. Stark, (1964). A Selection of Problems in the theory of numbers. Popular lectures in mathematics, Elsevier Ltd, Macmillan Company.

[2] Kenneth H. Rosen, (2011). Elementary number theory and its applications, 6th Edition, Monmouth University.

[3] Y. D. Sergeyev, (2013). Numerical Computations with Infinite and Infinitesimal Numbers: Theory and Applications, 1st Edition, Vol. I, Springer.URL http://wwwinfo.deis.unical.it/~yaro/DIS_book_Sergeyev.pdf

[4] M. T. Hamood, S. Boussakta, (2014). Efficient algorithms for computing the new Mersenne number transform, Digital Signal Processing 25. 280 – 288. doi: https://doi.org/10.1016/j.dsp.2013.10.018.

[5] C. W., (2006). Number theory: an introduction to mathematics, Vol. Part A-B, Springer.

[6] C. E. G. Pineda, S. M. García, (2011). Algunos tópicos en teoría de números: Números Mersenne, teorema Dirichlet, números Fermat, Scientia et Technica 2 (48). 185–190. doi: https://doi.org/10.22517/ 23447214.1279.9

[7] B. H. Gross, (2005). An elliptic curve test for Mersenne primes, Journal of Number Theory 110 (1). 114 – 119, Arnold Ross Memorial Issue. doi: https://doi.org/10.1016/j.jnt.2003.11.011.

[8] H. Helfgot, (2013). Mayor arcs for Goldbach problem, arXiv 14 (1) (2013) 1–79. URLhttps://arxiv.org/abs/1305.2897