scholarly journals Some properties of prime numbers of special form and Carmichael numbers

2016 ◽  
Vol 24 ◽  
pp. 41
Author(s):  
N.V. Kalashnіkova
Keyword(s):  
Special Form ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Carmichael Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

We study some properties of structure of the multiplicative group $Z^*_m$, in case when m is Mersenne prime, Fermat or Carmichael number. Using the results of these studies, we obtain properties of Mersenne primes, Fermat and Carmichael numbers.

scholarly journals Some New Notes on Mersenne Primes and Perfect Numbers

2020 ◽  
Vol 3 (1) ◽  
pp. 15
Author(s):  
Leomarich F Casinillo
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Mathematical Structure ◽  
Prime Numbers ◽  
Perfect Number ◽  
Positive Divisor ◽  
Triangular Numbers ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

<p>Mersenne primes are specific type of prime numbers that can be derived using the formula <img title="\large M_p=2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;M_p=2^{p}-1" alt="" />, where <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> is a prime number. A perfect number is a positive integer of the form <img title="\large P(p)=2^{p-1}(2^{p}-1)" src="https://latex.codecogs.com/gif.latex?\large&amp;space;P(p)=2^{p-1}(2^{p}-1)" alt="" /> where <img title="\large 2^{p}-1" src="https://latex.codecogs.com/gif.latex?\large&amp;space;2^{p}-1" alt="" /> is prime and <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" /> 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 <img title="\large p" src="https://latex.codecogs.com/gif.latex?\large&amp;space;p" alt="" />. 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.</p>

Pseudoperfect numbers with no small prime divisors

2009 ◽  
Vol 93 (528) ◽  
pp. 404-409
Author(s):  
Peter Shiu
Keyword(s):  
Difficult Problem ◽  
Prime Divisors ◽  
Perfect Number ◽  
Perfect Numbers ◽  
Mersenne Primes ◽  
Mersenne Prime

A perfect number is a number which is the sum of all its divisors except itself, the smallest such number being 6. By results due to Euclid and Euler, all the even perfect numbers are of the form 2P-1(2p - 1) where p and 2p - 1 are primes; the latter one is called a Mersenne prime. Whether there are infinitely many Mersenne primes is a notoriously difficult problem, as is the problem of whether there is an odd perfect number.

Investigating Prime Numbers and the Great Internet Mersenne Prime Search

2002 ◽  
Vol 8 (2) ◽  
pp. 70-76
Author(s):  
Jeffrey J. Wanko ◽  
Christine Hartley Venable
Keyword(s):  
Middle School ◽  
Middle School Students ◽  
Prime Number ◽  
Prime Numbers ◽  
School Students ◽  
Large Numbers ◽  
Mersenne Prime

Middle school students learn about patterns, formulas, and large numbers motivated by a search for the largest prime number. Activities included.

On the Schur-Zassenhaus theorem for groups of finite Morley rank

1992 ◽  
Vol 57 (4) ◽  
pp. 1469-1477 ◽  
Author(s):  
Alexandre V. Borovik ◽  
Ali Nesin
Keyword(s):  
Finite Group ◽  
Group Theory ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Hall Subgroup ◽  
Morley Rank ◽  
Finite Group Theory ◽  
Hall Subgroups ◽  
Finite Morley Rank ◽  
A Finite Group

The Schur-Zassenhaus Theorem is one of the fundamental theorems of finite group theory. Here is its statement:Fact 1.1 (Schur-Zassenhaus Theorem). Let G be a finite group and let N be a normal subgroup of G. Assume that the order ∣N∣ is relatively prime to the index [G:N]. Then N has a complement in G and any two complements of N are conjugate in G.The proof can be found in most standard books in group theory, e.g., in [S, Chapter 2, Theorem 8.10]. The original statement stipulated one of N or G/N to be solvable. Since then, the Feit-Thompson theorem [FT] has been proved and it forces either N or G/N to be solvable. (The analogous Feit-Thompson theorem for groups of finite Morley rank is a long standing open problem).The literal translation of the Schur-Zassenhaus theorem to the finite Morley rank context would state that in a group G of finite Morley rank a normal π-Hall subgroup (if it exists at all) has a complement and all the complements are conjugate to each other. (Recall that a group H is called a π-group, where π is a set of prime numbers, if elements of H have finite orders whose prime divisors are from π. Maximal π-subgroups of a group G are called π-Hall subgroups. They exist by Zorn's lemma. Since a normal π-subgroup of G is in all the π-Hall subgroups, if a group has a normal π-Hall subgroup then this subgroup is unique.)The second assertion of the Schur-Zassenhaus theorem about the conjugacy of complements is false in general. As a counterexample, consider the multiplicative group ℂ* of the complex number field ℂ and consider the p-Sylow for any prime p, or even the torsion part of ℂ*. Let H be this subgroup. H has a complement, but this complement is found by Zorn's Lemma (consider a maximal subgroup that intersects H trivially) and the use of Zorn's Lemma is essential. In fact, by Zorn's Lemma, any subgroup that has a trivial intersection with H can be extended to a complement of H. Since ℂ* is abelian, these complements cannot be conjugated to each other.

scholarly journals The average multiplicative order of a finitely generated subgroup of the rationals modulo primes

2016 ◽  
Vol 12 (08) ◽  
pp. 2147-2158
Author(s):  
Cihan Pehlivan
Keyword(s):  
Prime Number ◽  
Explicit Expression ◽  
Riemann Hypothesis ◽  
Multiplicative Group ◽  
Prime Numbers ◽  
Euler Product ◽  
Finitely Generated ◽  
Numerical Computations ◽  
Multiplicative Order ◽  
Reduction Group

Let [Formula: see text] be a finitely generated subgroup of [Formula: see text]. Let [Formula: see text] be a prime number for which the reduction group [Formula: see text] is a well-defined subgroup of the multiplicative group [Formula: see text], and denote the order of [Formula: see text] by [Formula: see text]. Assuming the Generalized Riemann Hypothesis, we study the average of [Formula: see text] and the average of the powers of [Formula: see text] as [Formula: see text] ranges over prime numbers. The problem was considered in the case of rank 1 by Pomerance and Kurlberg. In the case when [Formula: see text] contains only positive numbers, we give an explicit expression for the involved density in terms of an Euler product. We conclude with some numerical computations.

scholarly journals Constant-sized correlations are sufficient to self-test maximally entangled states with unbounded dimension

Quantum ◽  
2022 ◽  
Vol 6 ◽  
pp. 614
Author(s):  
Honghao Fu
Keyword(s):  
Multiplicative Group ◽  
Prime Numbers ◽  
Quarterly Journal ◽  
Entangled States ◽  
Entangled State ◽  
Local Dimension ◽  
Maximally Entangled State ◽  
Self Testing ◽  
Maximally Entangled States ◽  
Self Test

Let p be an odd prime and let r be the smallest generator of the multiplicative group Zp&#x2217;. We show that there exists a correlation of size &#x0398;(r2) that self-tests a maximally entangled state of local dimension p&#x2212;1. The construction of the correlation uses the embedding procedure proposed by Slofstra (Forum of Mathematics, Pi. (2019)). Since there are infinitely many prime numbers whose smallest multiplicative generator is in the set {2,3,5} (D.R. Heath-Brown The Quarterly Journal of Mathematics (1986) and M. Murty The Mathematical Intelligencer (1988)), our result implies that constant-sized correlations are sufficient for self-testing of maximally entangled states with unbounded local dimension.

Mersenne primes and solvable Sylow numbers

2016 ◽  
Vol 15 (09) ◽  
pp. 1650163
Author(s):  
Tian-Ze Li ◽  
Yan-Jun Liu
Keyword(s):  
Finite Group ◽  
Simple Group ◽  
Solvable Groups ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Nonabelian Simple Group ◽  
Mersenne Primes ◽  
A Finite Group ◽  
Mersenne Prime

Let [Formula: see text] be a prime. The Sylow [Formula: see text]-number of a finite group [Formula: see text], which is the number of Sylow [Formula: see text]-subgroups of [Formula: see text], is called solvable if its [Formula: see text]-part is congruent to [Formula: see text] modulo [Formula: see text] for any prime [Formula: see text]. P. Hall showed that solvable groups only have solvable Sylow numbers, and M. Hall showed that the Sylow [Formula: see text]-number of a finite group is the product of two kinds of factors: of prime powers [Formula: see text] with [Formula: see text] (mod [Formula: see text]) and of the number of Sylow [Formula: see text]-subgroups in certain finite simple groups (involved in [Formula: see text]). These classical results lead to the investigation of solvable Sylow numbers of finite simple groups. In this paper, we show that a finite nonabelian simple group has only solvable Sylow numbers if and only if it is isomorphic to [Formula: see text] for [Formula: see text] a Mersenne prime.

3. Prime-time mathematics

2020 ◽  
pp. 37-57
Author(s):  
Robin Wilson
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Prime Time ◽  
Leonhard Euler ◽  
Mersenne Primes ◽  
Fermat Primes

‘Prime-time mathematics’ explores prime numbers, which lie at the heart of number theory. Some primes cluster together and some are widely spread, while primes go on forever. The Sieve of Eratosthenes (3rd century BC) is an ancient method for identifying primes by iteratively marking the multiples of each prime as not prime. Every integer greater than 1 is either a prime number or can be written as a product of primes. Mersenne primes, named after French friar Marin de Mersenne, are prime numbers that are one less than a power of 2. Pierre de Fermat and Leonhard Euler were also prime number enthusiasts. The five Fermat primes are used in a problem from geometry.

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