Some New Notes on Mersenne Primes and Perfect Numbers

Leomarich F Casinillo

doi:10.31002/ijome.v3i1.2282

Title:

Some New Notes on Mersenne Primes and Perfect Numbers

Author:

Leomarich F Casinillo^(1*)

(1) Visayas State University,
(*) Corresponding Author
10.31002/ijome.v3i1.2282| Abstract views : 163 | PDF views : 0

Abstract

Mersenne primes are specific type of prime numbers that can be derived using the formula $M_p=2^{p}-1$ , where $p$ is a prime number. A perfect number is a positive integer of the form $P(p)=2^{p-1}(2^{p}-1)$ where $2^{p}-1$ is prime and $p$ is a Mersenne prime, and that can be written as the sum of its proper divisor, that is, a number that is half the sum of all of its positive divisor. In this note, some concepts relating to Mersenne primes and perfect numbers were revisited. Further, Mersenne primes and perfect numbers were evaluated using triangular numbers. This note also discussed how to partition perfect numbers into odd cubes for odd prime $p$ . Also, the formula that partition perfect numbers in terms of its proper divisors were constructed and determine the number of primes in the partition and discuss some concepts. The results of this study is useful to better understand the mathematical structure of Mersenne primes and perfect numbers.

Keywords

Mersenne primes; perfect numbers; triangular numbers

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References

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DOI: http://dx.doi.org/10.31002/ijome.v3i1.2282