REVISTA POLITÉCNICA, Vol. 16, Núm. 32

 

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.

 

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

 

 

 


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Revista Politécnica 
ISSN: 1900-2351
ISSN: 2256-5353 (En línea)

DOI:  10.33571/rpolitec

Editada por: Politécnico Colombiano Jaime Isaza Cadavid