Mersenne Numbers: consolidated results (original) (raw)
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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.
A Novel Deterministic Mersenne Prime Numbers Test: Aouessare-El Haddouchi-Essaaidi Primality Test
2015
There has been an increasing interest in prime numbers during the past three decades since the introduction of public-key cryptography owing to the large spread of internet and electronic banking. The largest prime number discovered so far, which is a Mersenne number, has 17,425,170 digits. However, the algorithmic complexity of Mersenne primes test is computationally very expensive. The best method presently known for Mersenne numbers primality testing is Lucas–Lehmer primality test. This paper presents a novel primality test for these numbers, namely, Aouessare-El Haddouchi-Essaaidi primality test, which largely outperforms Lucas-Lehmer test with its very low algorithmic complexity which allows performing much quicker tests with the other advantage of considerable memory requirements savings. Moreover, in the case of a composite number, where this test is negative, it is also possible to decompose the tested number into two factors whose product yields it. It is anticipated that t...
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.
Baghdad Science Journal
In this article, a new deterministic primality test for Mersenne primes is presented. It also includes a comparative study between well-known primality tests in order to identify the best test. Moreover, new modifications are suggested in order to eliminate pseudoprimes. The study covers random primes such as Mersenne primes and Proth primes. Finally, these tests are arranged from the best to the worst according to strength, speed, and effectiveness based on the results obtained through programs prepared and operated by Mathematica, and the results are presented through tables and graphs.
Analysis of Mersenne Primes using a Set of Prime Multiple Equations
Mersenne Primes are examined using a set of Primes Multiple Equations. These MPrimes are split into two groups and each is inspected by a different set of Prime Multiple equations. If any of the equations in the set is satisfied, then it's not a Mersenne Prime but a Prime Multiple, PM. Also, an interesting connection to Mersenne Primes is offered as a MPrime testing routine.
This paper attempts to speed-up the modular reduction as an independent step of modular multiplication, which is the central operation in public-key cryptosystems. Based on the properties of Mersenne and Quasi-Mersenne primes, we have described four distinct sets of moduli which are responsible for converting the single-precision multiplication prevalent in many of today's techniques into an addition operation and a few simple shift operations. We propose a novel revision to the Modified Barrett algorithm presented in [3]. With the backing of the special moduli sets, the proposed algorithm is shown to outperform (speed-wise) the Modified Barrett algorithm by 80% for operands of length 700 bits, the least speed-up being around 70% for smaller operands, in the range of around 100 bits.
Gaussian Mersenne and Eisenstein Mersenne primes
Mathematics of Computation, 2010
The Biquadratic Reciprocity Law is used to produce a deterministic primality test for Gaussian Mersenne norms which is analogous to the Lucas-Lehmer test for Mersenne numbers. It is shown that the proposed test could not have been obtained from the Quadratic Reciprocity Law and Proth's Theorem. Other properties of Gaussian Mersenne norms that contribute to the search for large primes are given. The Cubic Reciprocity Law is used to produce a primality test for Eisenstein Mersenne norms. The search for primes in both families (Gaussian Mersenne and Eisenstein Mersenne norms) was implemented in 2004 and ended in November 2005, when the largest primes, known at the time in each family, were found.
Some New Notes on Mersenne Primes and Perfect Numbers
Indonesian Journal of Mathematics Education, 2020
Mersenne primes are a specific type of prime number 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 a Mersenne prime and can be written as the sum of its proper divisor, that is, a number which is half the sum of all of its positive divisor. In this paper, some concepts relating to Mersenne primes and perfect numbers were revisited. Mersenne primes and perfect numbers were evaluated using triangular numbers. Further, this paper discussed how to partition perfect numbers into odd cubes for odd prime The formula that partition perfect numbers in terms of its proper divisors were developed. The results of this study are useful to understand the mathematical structures of Mersenne primes and perfect numbers.