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On a generalized sum of the Mersenne Numbers

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HAL Id: hal-01796401

https://hal.archives-ouvertes.fr/hal-01796401

Preprint submitted on 20 May 2018

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On a generalized sum of the Mersenne Numbers

Amelia Carolina Sparavigna

To cite this version:

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Zenodo 10.5281/zenodo.1250048

On a generalized sum of the Mersenne Numbers

Amelia Carolina Sparavigna Politecnico di Torino

Abstract Here we discuss the Mersenne numbers to give an example of a generalized sum. Using this sum, a recurrence relation is given.

Keywords Generalized sums, Mersenne numbers. DOI: 10.5281/zenodo.1250048

Written in Turin, on May 20, 2018

Of generalized sums of numbers, we have given some examples in previous works [1-3]. Here we propose the study of the Mersenne Numbers, using the same approach. About these numbers, a large literature exists (see for instance that given in [4]). The form of the numbers is that of a power of two minus 1. Among them we find the Mersenne primes. The numbers are named after Marin Mersenne (1588 – 1648), a French Minim friar, who studied them in the early 17th century.

Mersenne numbers are:

M

n

=2

n

−1

Let us consider them to give an example of generalized sum. We can start from the following calculus:

. Mm+n=2m+n−1

M

m+n

=2

m+n

−1=2

m

2

n

−1−2

m

+2

m

−2

n

+2

n

−1+1=2

m

(2

n

−1)−1+2

m

−2

n

+1+2

n

−1

M

m+n

=(2

m

−1)(2

n

−1)+2

m

−1+2

n

−1

Therefore, we can write the following generalized sum:

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Zenodo 10.5281/zenodo.1250048

This is a generalized sum that we find in the case of the multiplicative groups (for the use of multiplicative groups in statistics and statistical mechanics see [5,6]).

Using (1), for the Mersenne numbers we can imagine the following recursive relation:

M

n+1

=M

n

⊕M

1

=M

n

+M

1

+ M

n

M

1 That is:

2

n+1

−1=(2

n

−1)+(2

1

−1)+(2

n

−1)(2

1

−1)=2

n

+2

n+1

−2

n

−2+1=2

n+1

−1

The sum (1) is associative, so that:

Mm⊕Mn⊕Ml=Mm+Mn+Ml+MmMn+MnMl+MmMl+MmMnMl

We cannot have a group of the Mersenne numbers, without considering also the opposites of them, so that:

0=M

n

⊕Opposite(M

n

)

Therefore:

Opposite

(M

n

)=−

M

n

M

n

+1

=M

−n Explicitly:

Opposite

(2

n

−1)=− (

2

n

−1)

(2

n

−1)+1

=(−

2

n

+1)

2

n

=2

−n

−1

These numbers are the Mersenne numbers with a negative exponent. So we have: Mnnn=Mn⊕M−n=Mn+M−n+MnM−n

0=2

0

−1=(2

n

−1)+(2

−n

−1)+(2

n

−1)(2

−n

−1)=2

n

+2

−n

−2+2

n

2

−n

−2

−n

−2

n

+1=0

References

1. Sparavigna, A. C. (2018). Generalized Sums Based on Transcendental Functions. Zenodo . DOI: 10.5281/zenodo.1250020

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Zenodo 10.5281/zenodo.1250048

3. Sparavigna, A. C. (2018). ON THE ADDITIVE GROUP OF q-INTEGERS. Zenodo . DOI: 10.5281/zenodo.1245849

4. Weisstein, Eric W. "Mersenne Number." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/MersenneNumber.html

5. Sicuro, G., & Tempesta, P. (2016). Groups, information theory, and Einstein's likelihood principle. Phys. Rev. E 93, 040101(R).

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