scholarly journals A NOTE ON SIMULTANEOUS CONGRUENCES, II: MORDELL REVISED

2010 ◽  
Vol 88 (2) ◽  
pp. 261-275 ◽  
Author(s):  
TREVOR D. WOOLEY
Keyword(s):  
Prime Number ◽  
Natural Numbers ◽  
Simultaneous Congruences

AbstractWhen p is a prime number, and k1,…,kt are natural numbers with 1≤k1<k2<⋯<kt<p, we show that the simultaneous congruences ∑ t1xkji≡∑ t1ykjimod p (1≤j≤t) possess at most k1⋯ktpt solutions with 1≤xi,yi≤p (1≤i≤t). Analogous conclusions are provided when one or more of the exponents ki is negative.

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

Quelques résultats sur les équations axp + byp = cz2

2006 ◽  
Vol 58 (1) ◽  
pp. 115-153 ◽  
Author(s):  
W. Ivorra ◽  
A. Kraus
Keyword(s):  
Prime Number ◽  
Diophantine Equation ◽  
Hyperelliptic Curves ◽  
Rational Points ◽  
Modular Representations ◽  
Natural Numbers ◽  
Prime Divisors

AbstractLet p be a prime number ≥ 5 and a, b, c be non zero natural numbers. Using the works of K. Ribet and A. Wiles on the modular representations, we get new results about the description of the primitive solutions of the diophantine equation axp + byp = cz2, in case the product of the prime divisors of abc divides 2ℓ, with ℓ an odd prime number. For instance, under some conditions on a, b, c, we provide a constant f (a, b, c) such that there are no such solutions if p > f (a, b, c). In application, we obtain information concerning the ℚ-rational points of hyperelliptic curves given by the equation y2 = xp + d with d ∈ ℤ.

scholarly journals Some interesting tasks from the classical number theory

2020 ◽  
pp. 150-188
Author(s):  
Zurab Agdgomelashvili ◽  
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Prime Divisors ◽  
Problem Class

The article considers the following issues: – It’s of great interest for p and q primes to determine the number of those prime number divisors of a number 1 1 pq A p    that are less than p. With this purpose we have considered: Theorem 1. Let’s p and q are odd prime numbers and p  2q 1. Then from various individual divisors of the 1 1 pq A p    number, only one of them is less than p. A has at least two different simple divisors; Theorem 2. Let’s p and q are odd prime numbers and p  2q 1. Then all prime divisors of the number 1 1 pq A p    are greater than p; Theorem 3. Let’s q is an odd prime number, and p N \{1}, p]1;q] [q  2; 2q] , then each of the different prime divisors of the number 1 1 pq A p    taken separately is greater than p; Theorem 4. Let’s q is an odd prime number, and p{q1; 2q1}, then from different prime divisors of the number 1 1 pq A p    taken separately, only one of them is less than p. A has at least two different simple divisors. Task 1. Solve the equation 1 2 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 2. Solve the equation 1 3 1 z x y y    in the natural numbers x , y, z. In addition, y must be a prime number. Task 3. Solve the equation 1 1 z x y p y    where p{6; 7; 11; 13;} are the prime numbers, x, y  N and y is a prime number. There is a lema with which the problem class can be easily solved: Lemma ●. Let’s a, b, nN and (a,b) 1. Let’s prove that if an  0 (mod| ab|) , or bn  0 (mod| ab|) , then | ab|1. Let’s solve the equations ( – ) in natural x , y numbers: I. 2 z x y z z x y          ; VI. (x  y)xy  x y ; II. (x  y)z  (2x)z  yz ; VII. (x  y)xy  yx ; III. (x  y)z  (3x)z  yz ; VIII. (x  y) y  (x  y)x , (x  y) ; IV. ( y  x)x y  x y , (y  x) ; IX. (x  y)x y  xxy ; V. ( y  x)x y  yx , (y  x) ; X. (x  y)xy  (x  y)x , (y  x) . Theorem . If a, bN (a,b) 1, then each of the divisors (a2  ab  b2 ) will be similar. The concept of pseudofibonacci numbers is introduced and some of their properties are found.

scholarly journals Composition of infinitary structures

2015 ◽  
pp. 557-565
Author(s):  
Г.Г. Рябов ◽  
В.А. Серов
Keyword(s):  
Genetic Structure ◽  
Prime Number ◽  
Shortest Paths ◽  
Natural Numbers ◽  
Geometric Characteristics ◽  
Symbolic Matrix

Настоящая статья является продолжением рассмотрения полиморфных свойств троичных символьных матриц (TSM - Ternary Symbolic Matrix) над алфавитом $A=\{0,1,2\}$ как биекций кратчайших $k$-мерных путей между антиподальными вершинами ($skap$-путей) в $n$-кубе. Отображение TSM на структуру $k$-арного глобального дерева ($GTk$) определено как генетическое пространство $T(k)$ $skap$-путей. Автоморфизм TSM индуцирует нумерацию вершин $T(k)$ множеством натуральных чисел $\mathbb{N}$. С позиций такой структуры рассматриваются арифметическая геометрия $skap$-путей и свойства симметричности простых чисел относительно натуральных. В основу исследования симметричности простых предложены разностный таблоид DT (Difference Tabloid) и конструктивный метод оценки его наполнения как индикатора метрических отношений между натуральными и простыми числами. The infinitary structure of an $n$-cube, global $k$-ary trees, and natural numbers are considered as a single genetic structure. A number of geometric characteristics of the shortest paths in an $n$-cube are specified and the properties of prime number symmetry among the natural numbers are studied on the basis of this structure.

scholarly journals Quadratic nonresidues below the Burgess bound

2017 ◽  
Vol 13 (03) ◽  
pp. 751-759 ◽  
Author(s):  
William D. Banks ◽  
Victor Z. Guo
Keyword(s):  
Prime Number ◽  
Upper Bound ◽  
Legendre Symbol ◽  
Natural Numbers ◽  
Number Formula

For any odd prime number [Formula: see text], let [Formula: see text] be the Legendre symbol, and let [Formula: see text] be the sequence of positive nonresidues modulo [Formula: see text], i.e. [Formula: see text] for each [Formula: see text]. In 1957, Burgess showed that the upper bound [Formula: see text] holds for any fixed [Formula: see text]. In this paper, we prove that the stronger bound [Formula: see text] holds for all odd primes [Formula: see text] provided that [Formula: see text] where the implied constants are absolute. For fixed [Formula: see text], we also show that there is a number [Formula: see text] such that for all odd primes [Formula: see text], there are [Formula: see text] natural numbers [Formula: see text] with [Formula: see text] provided that [Formula: see text]

scholarly journals Distribuion of Leading Digits of Numbers II

2019 ◽  
Vol 14 (1) ◽  
pp. 19-42
Author(s):  
Yukio Ohkubo ◽  
Oto Strauch
Keyword(s):  
Distribution Function ◽  
Real Number ◽  
Prime Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Natural Numbers ◽  
Increasing Functions

AbstractIn this paper, we study the sequence (f (pn))n≥1,where pn is the nth prime number and f is a function of a class of slowly increasing functions including f (x)=logb xr and f (x)=logb(x log x)r,where b ≥ 2 is an integer and r> 0 is a real number. We give upper bounds of the discrepancy D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right) for a distribution function g and a sub-sequence (Ni)i≥1 of the natural numbers. Especially for f (x)= logb xr, we obtain the effective results for an upper bound of D_{{N_i}}^*\left( {f\left( {{p_n}} \right),g} \right).

scholarly journals Prime Numbers

2017 ◽  
Vol 5 (6) ◽  
pp. 41-66
Author(s):  
Mady Ndiaye
Keyword(s):  
Information Technology ◽  
Natural Number ◽  
Prime Number ◽  
Mathematical Reasoning ◽  
Prime Numbers ◽  
The Other ◽  
Natural Numbers ◽  
Mathematical Formulas ◽  
The Universe

A prime number is a natural number that has Just two divisors: one and itself. From antiquity until our time, scientists are researching mathematical reasoning to understand the prime numbers; eminent scholars had worked on this field before it is abandoned. Mathematicians considered the prime numbers like « building blocs in building natural numbers » and the field of mathematics the most difficult. Everything is about numbers, everything is about measure, The understanding of the natural numbers and more general the understanding of the numbers depend on the understanding of the prime numbers. This understanding of the prime will gives us greater ease to understand the other sciences. The prime numbers play a very important role for securing information technology hence promotion of the NTIC, Every year, there is a price for persons who will discover the biggest prime “it‟s the hunt for the big prime” This first part of this article about the prime numbers has taken a weight off the scientists „s shoulders by highlighting the universe of the prime numbers and has bring the problem of the prime numbers to an end. The mathematical formulas set out in this article allow us to determine all the biggest prime numbers compared to the capacity of our machines.

On the Distribution of Primes

2003 ◽  
Vol 96 (3) ◽  
pp. 198-200
Author(s):  
John M. Johnson
Keyword(s):  
Prime Number ◽  
Distribution Of Primes ◽  
Natural Numbers ◽  
Complete Understanding ◽  
Hall Of Fame

he distribution of primes throughout the natural numbers is a wonderful mystery that has always entertained mathematicians—professional and amateur, genius and ordinary—yet complete understanding has eluded their attempts. The names of those who have considered the problems discussed in this article and related problems form a “mathematics hall of fame” and span at least twenty-three centuries. This list includes Euclid, Euler, Fermat, Gauss, Legendre, and so on. This article highlights some of the well-known properties of primes and explores some properties that are not as wellknown but that are accessible to anyone who understands what a prime number is and who has a healthy amount of curiosity.

scholarly journals A REMARK ON PARTIAL SUMS INVOLVING THE MÖBIUS FUNCTION

2010 ◽  
Vol 81 (2) ◽  
pp. 343-349 ◽  
Author(s):  
TERENCE TAO
Keyword(s):  
Zeta Function ◽  
Prime Number ◽  
Elementary Proof ◽  
Prime Number Theorem ◽  
Möbius Function ◽  
Partial Sums ◽  
Natural Numbers ◽  
Nontrivial Zeros ◽  
Mobius Function ◽  
Zeros And Poles

AbstractLet 〈𝒫〉⊂N be a multiplicative subsemigroup of the natural numbers N={1,2,3,…} generated by an arbitrary set 𝒫 of primes (finite or infinite). We give an elementary proof that the partial sums ∑ n∈〈𝒫〉:n≤x(μ(n))/n are bounded in magnitude by 1. With the aid of the prime number theorem, we also show that these sums converge to ∏ p∈𝒫(1−(1/p)) (the case where 𝒫 is all the primes is a well-known observation of Landau). Interestingly, this convergence holds even in the presence of nontrivial zeros and poles of the associated zeta function ζ𝒫(s)≔∏ p∈𝒫(1−(1/ps))−1 on the line {Re(s)=1}. As equivalent forms of the first inequality, we have ∣∑ n≤x:(n,P)=1(μ(n))/n∣≤1, ∣∑ n∣N:n≤x(μ(n))/n∣≤1, and ∣∑ n≤x(μ(mn))/n∣≤1 for all m,x,N,P≥1.

scholarly journals Prime Numbers Calculation Formulas

2021 ◽  
pp. 1-6
Author(s):  
Ameha Tefera Tessema
Keyword(s):  
Computer Science ◽  
Prime Number ◽  
Modern Science ◽  
Prime Numbers ◽  
Jel Classification ◽  
Natural Numbers ◽  
Whole Numbers ◽  
Number Distribution

The application of prime numbers in modern science, especially in computer science, is very wide. Since prime numbers can only divisible by 1 and themselves, they are not factored any further like whole numbers. The problem to calculate all prime numbers using a formula posed for long periods. Though different formulas to calculate prime numbers were developed by Euler, Fermat, Mersenne and others, the formulas work for limited natural numbers and calculate limited prime numbers. JEL classification numbers: C02, C63, C69 Keywords: Prime numbers, Prime numbers formula, Prime number distribution, Prime number calculation.

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