Some New Notes on Mersenne Primes and Perfect Numbers (original) (raw)
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Some New Results on Perfect Numbers
2019
A perfect number is any positive integer nnn which satisfies sigma(n)=2n\sigma(n)=2nsigma(n)=2n, where sigma(n)\sigma(n)sigma(n) is the sum of positive factors of nnn. In this paper, using Dickson's approach to Euler theorem on even perfect numbers, we seek an answer to a Suryanarayana's question which demands whether every odd perfect number is of the form msigma(m)m\sigma(m)msigma(m). This paper thus gives the general form for perfect numbers with which it is possible to classify integers generally as perfect or non-perfect numbers. The form in this paper justifies Euclid-Euler's theorem, prunes Euler form for odd perfect numbers and allows us to give a remark concerning Suryanarayana's result that there is no odd super-perfect number of the form p2alphap^{2\alpha}p2alpha. We also recommend some problems for further study.
A COMPLETE CLASSIFICATION OF THE MERSENNE'S PRIMES AND ITS IMPLICATIONS FOR COMPUTING
2020
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.
Mersenne Variant Numbers and Integers investigated for Primality, Factorization
Mersenne Variant Numbers and Integers investigated for Primality, Factorization, 2008
Mersenne Variants are numbers of the form s^n +/- c where s,n >=2 and -s-1<=c<=s+1 and gcd(s,c)=1 which is a generalization of the type of Fermat and Mersenne Numbers. Here in this paper we give algorithms for primality of such numbers with mathematical proofs of correctness.
Sylvester: ushering in the modern era of research on odd perfect numbers
INTEGERS: ELECTRONIC JOURNAL OF …, 2003
In 1888, James Joseph Sylvester (1814-1897) published a series of papers that he hoped would pave the way for a general proof of the nonexistence of an odd perfect number (OPN). Seemingly unaware that more than fifty years earlier Benjamin Peirce had proved that an odd perfect number must have at least four distinct prime divisors, Sylvester began his fundamental assault on the problem by establishing the same result. Later that same year, he strengthened his conclusion to five. These findings would help to mark the beginning of the modern era of research on odd perfect numbers. Sylvester's bound stood as the best demonstrated until Gradstein improved it by one in 1925. Today, we know that the number of distinct prime divisors that an odd perfect number can have is at least eight. This was demonstrated by Chein in 1979 in his doctoral thesis. However, he published nothing of it. A complete proof consisting of almost 200 manuscript pages was given independently by Hagis. An outline of it appeared in 1980. What motivated Sylvester's sudden interest in odd perfect numbers? Moreover, we also ask what prompted this mathematician who was primarily noted for his work in algebra to periodically direct his attention to famous unsolved problems in number theory? The objective of this paper is to formulate a response to these questions, as well as to substantiate the assertion that much of the modern work done on the subject of odd perfect numbers has as it roots, the series of papers produced by Sylvester in 1888.
Formulating an Odd Perfect Number: An in Depth Case Study
A perfect number is a positive integer that is equals to the sum of its proper divisors. No one has ever found an odd perfect number in the field of Number Theory. This paper review discussed the history and the origin of Odd Perfect Numbers. The theorems and proofs are given and stated. This paper states the necessary conditions for the existence of odd perfect numbers. In addition, several related studies such as "Odd Near-Perfect Numbers" and "Deficient-Perfect Numbers". Formulating odd perfect numbers will have a significant contribution to other Mathematics conjectures. This paper compiles all the known information about the existence of an odd perfect number It also lists and explains the necessary theorems and lemmas needed for the study. The results and conclusions shows the ff: Odd Perfect Numbers has a lower bound of 10 1500 , The total number of prime factors/divisors of an odd perfect number is at least 101, and 10 8 is an appropriate lower bound for the largest prime factor of an odd perfect number and the second large stand third largest prime divisors must exceed 10000 and100 respectively. In summary, it found out that there is a chance for an odd perfect number to exist even if there is a very small possibility.
Commentationes Mathematicae, 2013
In this paper a modified form of perfect numbers called (p, q)+ perfect numbers and their properties with examples have been discussed. Further properties of σ + arithmetical function have been discussed and on its basis a modified form of perfect number called (p, q)+ super perfect numbers have been discussed. A modified form of perfect number called (p, 0)-perfect and their characterization has been studied. In the end of this paper almost super perfect numbers have been introduced.
No Odd Perfect Numbers Please! Revised Version 4.0
In this article we solve one of the oldest and celebrated problems in number theory, namely the existence or nonexistence of odd perfect numbers. We know there be no number of this type having less than 100 digits. A number is said to be perfect if it is the sum of its proper divisors. Euclid in his The Elements ninth book gives a formula for all even perfect numbers. We answer the question of whether there exists an odd perfect number in the negative by proving a theorem asserting that the existence of such a number would lead to contradictions (proof by reductio ad absurdum). Somewhat remarkably, perhaps, this result is proved using only elementary methods. Hence, the popular conjecture that odd perfect numbers do not exist, no matter how large these numbers might be, is confirmed to be correct. Thus, one of the oldest and celebrated questions in mathematics has now a definitive answer.
In this paper we show that Perfect Numbers are only "even" plus many other interesting relations about Mersenne"s prime. Furthermore, we describe also various equations, lemmas and theorems concerning the expression of a number as a sum of primes and the primitive divisors of Mersenne numbers. In conclusion, we show some possible mathematical connections between some equations regarding the arguments above mentioned and some sectors of string theory (p-adic and adelic strings and Ramanujan modular equation linked to the modes corresponding to the physical vibrations of the bosonic strings). REVISITED AND DEFINITIVE VERSION 12.11.2020