Exceptional Set in Waring–Goldbach Problem Involving Squares, Cubes and Sixth Powers

2021 ◽  
Vol 58 (1) ◽  
pp. 84-103
Author(s):  
Jinjiang Li ◽  
Min Zhang ◽  
Haonan Zhao
Keyword(s):  
Prime Numbers ◽  
Large Integer ◽  
Exceptional Set ◽  
Goldbach Problem ◽  
Positive Integers

Let N be a sufficiently large integer. In this paper, it is proved that, with at most O(N 119/270+s) exceptions, all even positive integers up to N can be represented in the form where p1, p2, p3, p4, p5, p6 are prime numbers.

scholarly journals Exceptional set in Waring–Goldbach problem for sums of one square and five cubes

2021 ◽  
Vol 7 (2) ◽  
pp. 2940-2955
Author(s):  
Jinjiang Li ◽  
◽  
Yiyang Pan ◽  
Ran Song ◽  
Min Zhang ◽  
...  
Keyword(s):  
Prime Numbers ◽  
Large Integer ◽  
Exceptional Set ◽  
Goldbach Problem ◽  
Positive Integers

<abstract><p>Let $ N $ be a sufficiently large integer. In this paper, it is proved that, with at most $ O\big(N^{4/9+\varepsilon}\big) $ exceptions, all even positive integers up to $ N $ can be represented in the form $ p_1^2+p_2^3+p_3^3+p_4^3+p_5^3+p_6^3 $, where $ p_1, p_2, p_3, p_4, p_5, p_6 $ are prime numbers.</p></abstract>

scholarly journals Vertex connectivity of the power graph of a finite cyclic group II

2019 ◽  
Vol 19 (02) ◽  
pp. 2050040 ◽  
Author(s):  
Sriparna Chattopadhyay ◽  
Kamal Lochan Patra ◽  
Binod Kumar Sahoo
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Prime Numbers ◽  
Vertex Connectivity ◽  
Induced Subgraph ◽  
Power Graph ◽  
Finite Cyclic Group ◽  
Minimum Number ◽  
Positive Integers ◽  
Appl Math

The power graph [Formula: see text] of a given finite group [Formula: see text] is the simple undirected graph whose vertices are the elements of [Formula: see text], in which two distinct vertices are adjacent if and only if one of them can be obtained as an integral power of the other. The vertex connectivity [Formula: see text] of [Formula: see text] is the minimum number of vertices which need to be removed from [Formula: see text] so that the induced subgraph of [Formula: see text] on the remaining vertices is disconnected or has only one vertex. For a positive integer [Formula: see text], let [Formula: see text] be the cyclic group of order [Formula: see text]. Suppose that the prime power decomposition of [Formula: see text] is given by [Formula: see text], where [Formula: see text], [Formula: see text] are positive integers and [Formula: see text] are prime numbers with [Formula: see text]. The vertex connectivity [Formula: see text] of [Formula: see text] is known for [Formula: see text], see [Panda and Krishna, On connectedness of power graphs of finite groups, J. Algebra Appl. 17(10) (2018) 1850184, 20 pp, Chattopadhyay, Patra and Sahoo, Vertex connectivity of the power graph of a finite cyclic group, to appear in Discr. Appl. Math., https://doi.org/10.1016/j.dam.2018.06.001]. In this paper, for [Formula: see text], we give a new upper bound for [Formula: see text] and determine [Formula: see text] when [Formula: see text]. We also determine [Formula: see text] when [Formula: see text] is a product of distinct prime numbers.

THE EXCEPTIONAL SET IN THE POLYNOMIAL GOLDBACH PROBLEM

2011 ◽  
Vol 07 (03) ◽  
pp. 579-591 ◽  
Author(s):  
PAUL POLLACK
Keyword(s):  
Asymptotic Formula ◽  
Finite Field ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Natural Numbers ◽  
Exceptional Set ◽  
Goldbach Problem

For each natural number N, let R(N) denote the number of representations of N as a sum of two primes. Hardy and Littlewood proposed a plausible asymptotic formula for R(2N) and showed, under the assumption of the Riemann Hypothesis for Dirichlet L-functions, that the formula holds "on average" in a certain sense. From this they deduced (under ERH) that all but Oϵ(x1/2+ϵ) of the even natural numbers in [1, x] can be written as a sum of two primes. We generalize their results to the setting of polynomials over a finite field. Owing to Weil's Riemann Hypothesis, our results are unconditional.

scholarly journals Asymptotic counting theorems for primitive juggling patterns

2019 ◽  
Vol 15 (05) ◽  
pp. 1037-1050
Author(s):  
Erik R. Tou
Keyword(s):  
Prime Number ◽  
Generating Functions ◽  
Arbitrary Number ◽  
Mathematical Theory ◽  
Prime Number Theorem ◽  
Prime Numbers ◽  
Theoretical Framework ◽  
Positive Integers ◽  
Infinite Set ◽  
Fixed Period

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.

On a Diophantine equation with prime numbers

2019 ◽  
Vol 15 (08) ◽  
pp. 1601-1616
Author(s):  
Sanhua Li
Keyword(s):  
Real Number ◽  
Diophantine Equation ◽  
Prime Numbers ◽  
Large Integer ◽  
The Real ◽  
Number Formula

Let [Formula: see text] denote the integral part of the real number [Formula: see text]. In this paper, it is proved that for [Formula: see text], the Diophantine equation [Formula: see text] is solvable in prime variables [Formula: see text] for sufficiently large integer [Formula: see text].

scholarly journals Partition-theoretic formulas for arithmetic densities, II

2021 ◽  
Vol Volume 43 - Special... ◽  
Author(s):  
Ken Ono ◽  
Robert Schneider ◽  
Ian Wagner
Keyword(s):  
Dirichlet Series ◽  
First Principles ◽  
Prime Numbers ◽  
Integer Partitions ◽  
Root Of Unity ◽  
International Audience ◽  
Positive Integers ◽  
Limiting Values ◽  
Original Theorem ◽  
Do So

International audience In earlier work generalizing a 1977 theorem of Alladi, the authors proved a partition-theoretic formula to compute arithmetic densities of certain subsets of the positive integers N as limiting values of q-series as q → ζ a root of unity (instead of using the usual Dirichlet series to compute densities), replacing multiplicative structures of N by analogous structures in the integer partitions P. In recent work, Wang obtains a wide generalization of Alladi's original theorem, in which arithmetic densities of subsets of prime numbers are computed as values of Dirichlet series arising from Dirichlet convolutions. Here the authors prove that Wang's extension has a partition-theoretic analogue as well, yielding new q-series density formulas for any subset of N. To do so, we outline a theory of q-series density calculations from first principles, based on a statistic we call the "q-density" of a given subset. This theory in turn yields infinite families of further formulas for arithmetic densities.

scholarly journals ON THE WARING–GOLDBACH PROBLEM FOR CUBES

2009 ◽  
Vol 51 (3) ◽  
pp. 703-712 ◽  
Author(s):  
JÖRG BRÜDERN ◽  
KOICHI KAWADA
Keyword(s):  
Natural Number ◽  
Large Integer ◽  
Natural Numbers ◽  
Goldbach Problem ◽  
Almost All

AbstractWe prove that almost all natural numbers satisfying certain necessary congruence conditions can be written as the sum of two cubes of primes and two cubes of P2-numbers, where, as usual, we call a natural number a P2-number when it is a prime or the product of two primes. From this result we also deduce that every sufficiently large integer can be written as the sum of eight cubes of P2-numbers.

On a Waring-Goldbach problem involving squares and cubes

2019 ◽  
Vol 69 (6) ◽  
pp. 1249-1262
Author(s):  
Yuhui Liu
Keyword(s):  
Asymptotic Formula ◽  
Natural Number ◽  
Goldbach Problem ◽  
Positive Integers

AbstractLet R(n) denote the number of representations of a natural number n as the sum of two squares and four cubes of primes. In this paper, it is proved that the anticipated asymptotic formula for R(n) fails for at most $\begin{array}{}O(N^{\frac{1}{4} + \varepsilon})\end{array}$ positive integers not exceeding N.

Hua’s theorem on sums of five prime squares in arithmetic progressions

2008 ◽  
Vol 45 (1) ◽  
pp. 29-66
Author(s):  
Claus Bauer
Keyword(s):  
Prime Numbers ◽  
Arithmetic Progressions ◽  
Prime Squares ◽  
Positive Integers

It is proved that for a given integer N and for all but ⪡ (log N ) B prime numbers k ≦ N5/96 − ε the following is true: For any positive integers bi , i ∈ {1, 2, 3, 4, 5}, ( bi , k ) = 1 that satisfy N ≡ b12 + b22 + b32 + b42 + b52 (mod k ), N can be written as N = p12 + p22 + p32 + p42 + p52 , where the pi , i ∈ {1, 2, 3, 4, 5} are prime numbers that satisfy pi ≡ bi (mod k ).

scholarly journals Solving Program Sketches with Large Integer Values

2020 ◽  
pp. 572-598
Author(s):  
Rong Pan ◽  
Qinheping Hu ◽  
Rishabh Singh ◽  
Loris D’Antoni
Keyword(s):  
Number Theory ◽  
Chinese Remainder Theorem ◽  
Program Synthesis ◽  
Prime Numbers ◽  
Large Integer ◽  
Complex Program

AbstractProgram sketching is a program synthesis paradigm in which the programmer provides a partial program with holes and assertions. The goal of the synthesizer is to automatically find integer values for the holes so that the resulting program satisfies the assertions. The most popular sketching tool, Sketch, can efficiently solve complex program sketches, but uses an integer encoding that often performs poorly if the sketched program manipulates large integer values. In this paper, we propose a new solving technique that allows Sketch to handle large integer values while retaining its integer encoding. Our technique uses a result from number theory, the Chinese Remainder Theorem, to rewrite program sketches to only track the remainders of certain variable values with respect to several prime numbers. We prove that our transformation is sound and the encoding of the resulting programs are exponentially more succinct than existing Sketch encodings. We evaluate our technique on a variety of benchmarks manipulating large integer values. Our technique provides speedups against both existing Sketch solvers and can solve benchmarks that existing Sketch solvers cannot handle.

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