Consecutive primes in tuples

William D. Banks; Tristan Freiberg; Caroline L. Turnage-Butterbaugh

Displaying similar documents to “Consecutive primes in tuples”

Associated primes, integral closures and ideal topologies

Reza Naghipour (2006)

Colloquium Mathematicae

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Let ⊆ be ideals of a Noetherian ring R, and let N be a non-zero finitely generated R-module. The set Q̅*(,N) of quintasymptotic primes of with respect to N was originally introduced by McAdam. Also, it has been shown by Naghipour and Schenzel that the set $A *_{a} (, N) : = ⋃_{n \geq 1} A s s_{R} R / {(ⁿ)}_{a}^{(N)}$ of associated primes is finite. The purpose of this paper is to show that the topology on N defined by ${{(ⁿ)}_{a}^{(N)} :_{R} ⟨ ⟩}_{n \geq 1}$ is finer than the topology defined by ${(ⁿ)}_{a}^{(N)}_{n \geq 1}$ if and only if $A *_{a} (, N)$ is disjoint from the quintasymptotic primes of with respect to N. Moreover,...

On a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let $p_{1}, p_{2}, \dots$ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if $k \geq 5$ , then $(p_{k + 1} - 1)! ∣ (\frac{1}{2} (p_{k + 1} - 1))! p_{k}!,$ which improves a previous result of the second author.

Goldbach’s problem with primes in arithmetic progressions and in short intervals

Karin Halupczok (2013)

Journal de Théorie des Nombres de Bordeaux

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Some mean value theorems in the style of Bombieri-Vinogradov’s theorem are discussed. They concern binary and ternary additive problems with primes in arithmetic progressions and short intervals. Nontrivial estimates for some of these mean values are given. As application inter alia, we show that for large odd $n \neg \equiv 1 (6)$ , Goldbach’s ternary problem $n = p_{1} + p_{2} + p_{3}$ is solvable with primes $p_{1}, p_{2}$ in short intervals $p_{i} \in [X_{i}, X_{i} + Y]$ with $X_{i}^{θ_{i}} = Y$ , $i = 1, 2$ , and $θ_{1}, θ_{2} \geq 0.933$ such that $(p_{1} + 2) (p_{2} + 2)$ has at most $9$ prime factors.

Sums of positive density subsets of the primes

Kaisa Matomäki (2013)

Acta Arithmetica

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We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1 - o (1)) α / (e^{γ} l o g l o g (1 / β))$ , which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ *_{m}$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B \subseteq ℤ *_{m}$ of densities α...

Non-Wieferich primes in number fields and $a b c$ -conjecture

Srinivas Kotyada, Subramani Muthukrishnan (2018)

Czechoslovak Mathematical Journal

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Let $K / ℚ$ be an algebraic number field of class number one and let $𝒪_{K}$ be its ring of integers. We show that there are infinitely many non-Wieferich primes with respect to certain units in $𝒪_{K}$ under the assumption of the $a b c$ -conjecture for number fields.

On a recursive formula for the sequence of primes and applications to the twin prime problem

Giovanni Fiorito (2006)

Bollettino dell'Unione Matematica Italiana

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In this paper we give a recursive formula for the sequence of primes $\{p_{n}\}$ and apply it to find a necessary and sufficient condition in order that a prime number $p_{n+1}$ is equal to $p_{n}+2$ . Applications of previous results are given to evaluate the probability that $p_{n+1}$ is of the form $p_{n}+2$ ; moreover we prove that the limit of this probability is equal to zero as $n$ goes to $\infty$ . Finally, for every prime $p_{n}$ we construct a sequence whose terms that are in the interval $[p_{n}^{2}-2,p_{n+1}^{2}-2[$ are the first terms of two twin primes. This...

Density of solutions to quadratic congruences

Neha Prabhu (2017)

Czechoslovak Mathematical Journal

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A classical result in number theory is Dirichlet’s theorem on the density of primes in an arithmetic progression. We prove a similar result for numbers with exactly $k$ prime factors for $k > 1$ . Building upon a proof by E. M. Wright in 1954, we compute the natural density of such numbers where each prime satisfies a congruence condition. As an application, we obtain the density of squarefree $n \leq x$ with $k$ prime factors such that a fixed quadratic equation has exactly $2^{k}$ solutions modulo $n$ . ...

On the least almost-prime in arithmetic progression

Jinjiang Li, Min Zhang, Yingchun Cai (2023)

Czechoslovak Mathematical Journal

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Let $𝒫_{r}$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. Suppose that $a$ and $q$ are positive integers satisfying $(a, q) = 1$ . Denote by $𝒫_{2} (a, q)$ the least almost-prime $𝒫_{2}$ which satisfies $𝒫_{2} \equiv a (mod q)$ . It is proved that for sufficiently large $q$ , there holds $𝒫_{2} (a, q) ≪ q^{1.8345} .$ This result constitutes an improvement upon that of Iwaniec (1982), who obtained the same conclusion, but for the range $1.845$ in place of $1.8345$ .

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

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Given an integer $n \geq 3$ , let $u_{1}, ..., u_{n}$ be pairwise coprime integers $\geq 2$ , $𝒟$ a family of nonempty proper subsets of ${1, ..., n}$ with “enough” elements, and $ε$ a function $𝒟 \to {\pm 1}$ . Does there exist at least one prime $q$ such that $q$ divides $\prod_{i \in I} u_{i} - ε (I)$ for some $I \in 𝒟$ , but it does not divide $u_{1} \dots u_{n}$ ? We answer this question in the positive when the $u_{i}$ are prime powers and $ε$ and $𝒟$ are subjected to certain restrictions. We use the result to prove that, if $ε_{0} \in {\pm 1}$ and $A$ is a set of three or more primes that contains all prime divisors of any...

On the least almost-prime in arithmetic progressions

Liuying Wu (2024)

Czechoslovak Mathematical Journal

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Let $𝒫_{2}$ denote a positive integer with at most $2$ prime factors, counted according to multiplicity. For integers $a$ , $q$ such that $(a, q) = 1$ , let $𝒫_{2} (q, a)$ denote the least $𝒫_{2}$ in the arithmetic progression ${n q + a}_{n = 1}^{\infty}$ . It is proved that for sufficiently large $q$ , we have $𝒫_{2} (q, a) ≪ q^{1.825} .$ This result constitutes an improvement upon that of J. Li, M. Zhang and Y. Cai (2023), who obtained $𝒫_{2} (q, a) ≪ q^{1.8345} .$

Infinitude of Wilson primes for $_{q} [t]$

Jim Sauerberg, Linghsueh Shu, Dinesh S. Thakur, George Todd (2013)

Acta Arithmetica

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