Set Of Prime Numbers Research Articles

Continued fractions of arithmetic sequences of quadratics

Let x be a quadratic irrational and let P be the set of prime numbers. We show the existence of an infinite set S⊂P such that the statistics of the period of the continued fraction expansions along the sequence px:p∈S approach the ‘normal’ statistics given by the Gauss–Kuzmin measure. Under the generalized Riemann hypothesis, we prove that there exist full density subsets S⊂P and T⊂N satisfying the same assertion. We give a rate of convergence in all cases.

Asymptotic counting theorems for primitive juggling patterns

The mathematics of juggling emerged after the development of siteswap notation in the 1980s. Consequently, much work was done to establish a mathematical theory that describes and enumerates the patterns that a juggler can (or would want to) execute. More recently, mathematicians have provided a broader picture of juggling sequences as an infinite set possessing properties similar to the set of positive integers. This theoretical framework moves beyond the physical possibilities of juggling and instead seeks more general mathematical results, such as an enumeration of juggling patterns with a fixed period and arbitrary number of balls. One problem unresolved until now is the enumeration of primitive juggling sequences, those fundamental juggling patterns that are analogous to the set of prime numbers. By applying analytic techniques to previously-known generating functions, we give asymptotic counting theorems for primitive juggling sequences, much as the prime number theorem gives asymptotic counts for the prime positive integers.

The number of linear factors of supersingular polynomials and sporadic simple groups

The set of prime numbers p such that the supersingular j-invariants in characteristic p are all contained in the prime field is finite. And it is well known that this set of primes coincides with the set of prime divisors of the order of the Monster simple group. In this paper, we will present analogous coincidences of supersingular invariants in level 2 and 3 and the orders of the Baby monster group and the Fischer's group. The proof uses a connection between the number of supersingular invariants and class numbers of imaginary quadratic fields.

Somme translatée sur des suites primitives et la conjecture d'Erdös

In this note, we construct a set S of primitive sequences such that, for any real number x≥81, we get∑a∈A1a(log⁡a+x)>∑p∈P1p(log⁡p+x), A∈S where P denotes the set of prime numbers.

On the least common multiple of several random integers

Let Ln(k) denote the least common multiple of k independent random integers uniformly chosen in {1,2,…,n}. In this article, using a purely probabilistic approach, we derive a criterion for the convergence in distribution as n→∞ of f(Ln(k))nrk for a wide class of multiplicative arithmetic functions f with polynomial growth r∈R. Furthermore, we identify the limit as an infinite product of independent random variables indexed by the set of prime numbers. Along the way, we compute the generating function of a trimmed sum of independent geometric laws, occurring in the above infinite product. This generating function is rational; we relate it to the generating function of a certain max-type Diophantine equation, of which we solve a generalized version. Our results extend theorems by Erdős and Wintner (1939), Fernández and Fernández (2013) and Hilberdink and Tóth (2016).

The Erdős conjecture for primitive sets

A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that the sum of 1 / ( a log ⁡ a ) 1/(a\log a) for a a running over a primitive set A A is universally bounded over all choices for A A . In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts and show a connection to certain prime number “races” such as the race between π ( x ) \pi (x) and l i ( x ) \mathrm {li}(x) .

ANALYSIS OF THE STRUCTURES OF ALGEBRAIC DYNAMICAL SYSTEMS BASED ON A COMPUTER SOLUTION OF THE GENERALIZED ARTIN HYPOTHESIS

The article considers the generalized Artin's hypothesis. The analysis of algebraic dynamical systems on the set of prime numbers is given. The properties of dynamic algebraic systems are studied.Based on computer modeling, a solution of Artin’s generalized hypothesis was constructed. A classification of prime numbers for any natural number a 1 is constructed. The properties of classes of prime numbers are investigated. A method of structural analysis of algebraic dynamical systems with close values of generalized Artin constants was developed. It is established that for any a 1 each class has a probability measure, and the sum of the measures of the classes tends to unity.

РОЗПАРАЛЕЛЕННЯ ОБЧИСЛЕННЯ КАНОНІЧНОГО РОЗКЛАДУ ЧИСЛА НА МНОЖНИКИ

The cоmputation of the canonical factorization of a number using the modified trial divisions method has been considered. The performing operations of the division a number of factorization into prime numbers for the testing on a repetition factor demands a respective loss of the execute time in modern computer systems. To reduce them, the presentation of the number of factorization in the binary form is used for the process of analysis on repetition factors. The residuals of weighting coefficient are defined for each digit of the binary representation the number of factorization, which is equal to one. The obtained values ​​of the residuals are accumulated and then the accumulated value is checked for the equality with the corresponding value from the set of prime numbers. In case of equality, we obtain an element of canonical factorization and again check the degrees of this element for a repetition factor. Otherwise, we proceed to the next larger prime number for further checking for a repetition factor. The independence of the subtasks to perform the check of the binary representation of a number on divisibility by prime numbers makes it possible to parallelize the execution of the factorization of a number in multi-core microprocessors of computer systems. Among the levels of the parallelism can be consistently identified: the definition of residual weighting coefficients, the accumulation of residuals for bits equel to one of the binary representation of the number of factorization, the checking for a repetition factor from the sets of primes. Software implementation in C++, according to the algorithm, schedules the computations in multithreads, using a pool of threads. In the algorithm for parallelizing the computations of the canonical factorization, depending on the entered value of the expansion number, the corresponding value of the number of primes with their powers is determined and is evenly distributed between the streams to perform an analysis of divisibility. As a result, the dependence of the run time the computation of the factorization of a number from the number of threads is defined in multi-core processors of Intel Core i3/i5/i7. For the each computer system exist the optimal number of the threads, which supports the minimal time of the canonical factorization of a number on the prime numbers.

О некоторых характеристиках множества простых чисел

The set of prime numbers p ≥ 5 is divided into two nonoverlapping subset P1 = {6k1  1}, P2 = {6k2 + 1}, where ki ⋲ A (i = 1,2). Subsets A1, A2 of natural numbers is defined by differences Ai = N\Bi, where B1, B2 are subset {j1}, {j2} defining subsets {6j1 – 1}, {6j2 + 1} of odd composite numbers. In [1] is proved two theorems permitting easily find by means of arithmetic progression subset Bi for ji  a ⋲ N. The tables of numbers ki for a = 500 are defined and some characteristic of subsets P1, P2 are given.

Normal numbers with digit dependencies

We give metric theorems for the property of Borel normality for real numbers under the assumption of digit dependencies in their expansion in a given integer base. We quantify precisely how much digit dependence can be allowed such that almost all real numbers are normal. Our theorem states that almost all real numbers are normal when at least slightly more than log ⁡ log ⁡ n \log \log n consecutive digits with indices starting at position n n are independent. As the main application, we consider the Toeplitz set T P T_P , which is the set of all sequences a 1 a 2 … a_1a_2 \ldots of symbols from { 0 , … , b − 1 } \{0, \ldots , b-1\} such that a n a_n is equal to a p n a_{pn} for every p p in P P and n = 1 , 2 , … n=1,2,\ldots . Here b b is an integer base and P P is a finite set of prime numbers. We show that almost every real number whose base b b expansion is in T P T_P is normal to base b b . In the case when P P is the singleton set { 2 } \{2\} we prove that more is true: almost every real number whose base b b expansion is in T P T_P is normal to all integer bases. We also consider the Toeplitz transform which maps the set of all sequences to the set T P T_P , and we characterize the normal sequences whose Toeplitz transform is normal as well.

Additively unique sets of prime numbers

Spiro proved that the identity function is the only multiplicative function with [Formula: see text] for some prime [Formula: see text] and [Formula: see text] for all prime [Formula: see text] and [Formula: see text]. We determine the sets [Formula: see text] of primes for which restricting our condition to [Formula: see text] for all [Formula: see text] still implies that [Formula: see text] is the identity function. We prove that [Formula: see text] satisfies these conditions if and only if [Formula: see text] contains every prime that is not the larger element of a twin prime pair and [Formula: see text] contains [Formula: see text] or [Formula: see text].

Prime Residue Class of Uniform Charges on the Integers

There is a probability charge on the power set of the integers that gives probability 1 / p to every residue class modulo a prime p. There exists such a charge that gives probability w to the set of prime numbers iff $$w \in [0,1/2]$$. Similarly, there is such a charge that gives probability x to a residue class modulo c, where c is composite, iff $$x \in [0,1/y]$$, where y is the largest prime factor of c.

On regular bipartite divisor graph for the set of irreducible character degrees

Given a finite group $G$, the \textit{bipartite divisor graph}, denoted by $B(G)$, for its irreducible character degrees is the bipartite graph with bipartition consisting of $cd(G)^{*}$, where $cd(G)^{*}$ denotes the nonidentity irreducible character degrees of $G$ and the $\rho(G)$ which is the set of prime numbers that divide these degrees, and with $\{p,n\}$ being an edge if $\gcd(p,n)\neq 1$. In [Bipartite divisor graph for the set of irreducible character degress, Int. J. Group Theory, 2017], the author considered the cases where $B(G)$ is a path or a cycle and discussed some properties of $G$. In particular she proved that $B(G)$ is a cycle if and only if $G$ is solvable and $B(G)$ is either a cycle of length four or six. Inspired by $2$-regularity of cycles, in this paper we consider the case where $B(G)$ is an $n$-regular graph for $n\in\{1,2,3\}$. In particular we prove that there is no solvable group whose bipartite divisor graph is $C_{4}+C_{6}$.

On Brauer p-dimensions and absolute Brauer p-dimensions of Henselian fields

This paper determines the Brauer p-dimension Brdp(K) and the absolute Brauer p-dimension abrdp(K) of a Henselian valued field (K,v), for a prime p≠char(Kˆ), under restrictions on the residue field Kˆ, such as the condition abrdp(Kˆ)=0. It describes the set Σ0 of sequences abrdp(E),Brdp(E), p∈P, where P is the set of prime numbers and E runs across the class of Henselian fields with char(Eˆ)=0 and a projective absolute Galois group GEˆ. Specifically, Σ0 contains a sequence ap,bp∈N∪{0,∞}, p∈P, whenever a2≤2b2 and ap≥bp, for each p. Similar results are obtained in characteristic q>0.

Test sets for polynomials: n-universal subsets and Newton sequences

Let E be a subset of an integral domain D with quotient field K. A subset S of E is said to be an n-universal subset of E if every integer-valued polynomial f(X)∈K[X] on S (that is, such that f(S)⊆D), with degree at most n, is integer-valued on E (that is, f(E)⊆D). A sequence a0,…,an of elements of E is said to be a Newton sequence of E of length n if, for each k≤n, the subset {a0,…,ak} is a k-universal subset of E. Our main results concern the case where D is a Dedekind domain, where both notions are strongly linked to p-orderings, as introduced by Bhargava. We extend and strengthen previous studies by Volkov, Petrov, Byszewski, Fra̧czyk, and Szumowicz that concerned only the case where E=D. In this case, but also if E is an ideal of D, or if E is the set of prime numbers >n+1 (in D=Z), we prove the existence of sequences in E of which n+2 consecutive terms always form an n-universal subset of E.

ПРОЦЕДУРА РЕШЕНИЯ БИНАРНОЙ ПРОБЛЕМЫ ГОЛЬДБАХА - ЭЙЛЕРА НА БАЗЕ ЧИСЕЛ СПЕЦИАЛЬНОГО ТИПА

Since the elements are closed, the set Θ = {6λ ± 1; λ ∈ N }, is a semigroup with respect to the operation of multiplication. The paper focuses on presenting even numbers ζ > 8 in the form of sums of two elements: θ1 = 6λ1 ± 1 and θ2 = 6λ2 ± 1 from the set Θ. Any even number ζ > 8 is comparable with one of the numbers m = (0, 2, -2), according to (mod 6). According to the remnants listed m , even numbers ζ > 8 are divided into 3 types. Each type has its own way of presenting sums in the form of two elements from the set Θ. For any even number ζ > 8 on the segment [1 ÷ ν] there is always at least a pair of numbers (λ1, λ2) ∈ [1 ÷ ν], that both are elements from the union of sets: the parameters of the prime numbers (twins) and the parameters (composite and prime) of numbers Θ. A variant of the solution of Goldbach - Euler conjecture for even numbers ζ > 8 is given on the set of primes P. Goldbach - Euler conjecture is also solvable in the set of prime numbers (twins), if the parameters of numbers θ1 and θ2, i.e. λ1 and λ2 belong to the set N \ FN , where FN is the set of parameters of the composite numbers Θ on the segment [1 ÷ ν].

Prime-Residue-Class of Uniform Charges on the Integers

There is a probability charge on the power set of the integers that gives probability 1/p to every residue class modulo a prime p. There exists such a charge that gives probability w to the set of prime numbers iff w ∈ [0, 1/2]. Similarly, there is such a charge that gives probability x to the residue class modulo c, where c is composite, iff x ∈ [0, 1/y], where y is the largest prime factor of c.

On complete Hall $${\mathfrak {F}}$$-sets of subgroups of finite groups and $${\fancyscript{Z}}$$-permutability

Let $${\mathfrak {F}}$$ be a class of finite groups and let G be a finite group. Assume that $$\sigma = \{\sigma _i : i \in I\}$$ is a partition of the set of prime numbers $${\mathbb {P}}$$ . A set $${\fancyscript{Z}}$$ of subgroups of G is called a complete Hall $${\mathfrak {F}}$$ -set of subgroups of G of type $$\sigma $$ if (i) for every $$W \in {\fancyscript{Z}}$$ , $$W \in {\mathfrak {F}}$$ and W is a Hall $$\sigma _i$$ -subgroup of G for some $$i \in I$$ and if (ii) for every $$\sigma _i \in \sigma $$ , $${\fancyscript{Z}}$$ contains exactly one and only one Hall $$\sigma _i$$ -subgroup of G. A subgroup H of G is said to be $${\fancyscript{Z}}$$ -permutable in G if H permutes with every member of $${\fancyscript{Z}}$$ . We investigate the influence of $${\fancyscript{Z}}$$ -permutable subgroups on the structure of finite groups. Also, we study some properties of $${\fancyscript{Z}}$$ -permutable subgroups. Several results from the literature are improved and generalized.

Partially ordered sets of non-trivial nilpotent π-subgroups II

Let G be a finite group, and π a set of prime numbers dividing the order of G. Denote by Nπ(G) and Lπ(G) respectively the totality of non-trivial nilpotent π-subgroups of G, and that of all subgroups U in Nπ(G) such that OπZNG(U)≤U. In this paper, we study homotopy equivalences related to those two posets which are known to have the same homotopy type. As an application, we deal with homology Hn(Nπ(G)) of the associated order complex by making use of Mayer–Vietoris sequences. Furthermore, we provide an algorithm for determining Lπ(GL(n,pe)) where p∉π. The determination of this is eventually reduced to that of irreducible subgroups of GL(n,pe).

THE DIOPHANTINE PROBLEM FOR ADDITION AND DIVISIBILITY OVER SUBRINGS OF THE RATIONALS

AbstractIt is shown that the positive existential theory of the structure (ℤ[S−1]; =, 0, 1, + , |), where S is a nonempty finite set of prime numbers, is undecidable. This result should be put in contrast with the fact that the positive existential theory of (ℤ; =, 0, 1, + |) is decidable.