Displaying similar documents to “The binary Goldbach conjecture with primes in arithmetic progressions with large modulus”

Consecutive primes in tuples

William D. Banks, Tristan Freiberg, Caroline L. Turnage-Butterbaugh (2015)

Acta Arithmetica

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In a stunning new advance towards the Prime k-Tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for an admissible k-tuple ( x ) = g x + h j j = 1 k of linear forms in ℤ[x], the set ( n ) = g n + h j j = 1 k contains at least m primes for infinitely many n ∈ ℕ. In this note, we deduce that ( n ) = g n + h j j = 1 k contains at least m consecutive primes for infinitely many n ∈ ℕ. We answer an old question of Erdős and Turán by producing strings of m + 1 consecutive primes whose successive gaps δ 1 , . . . , δ m form an increasing...

On the Brun-Titchmarsh theorem

James Maynard (2013)

Acta Arithmetica

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The Brun-Titchmarsh theorem shows that the number of primes which are less than x and congruent to a modulo q is less than (C+o(1))x/(ϕ(q)logx) for some value C depending on logx/logq. Different authors have provided different estimates for C in different ranges for logx/logq, all of which give C>2 when logx/logq is bounded. We show that one can take C=2 provided that logx/logq ≥ 8 and q is sufficiently large. Moreover, we also produce a lower bound of size x / ( q 1 / 2 ϕ ( q ) ) when logx/logq ≥ 8 and...

Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

Zhao Feng (2017)

Open Mathematics

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In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i.e., [...] N=p13+…+pj3 N = p 1 3 + ... + p j 3 with [...] |pi−(N/j)1/3|≤N1/3−δ+ε(1≤i≤j), | p i - ( N / j ) 1 / 3 | ≤ N 1 / 3 - δ + ε ( 1 ≤ i ≤ j ) , for some [...] 0<δ≤190. 0 δ ≤ 1 90 . Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.

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 ≤ x with k prime factors such that a fixed quadratic equation has exactly 2 k solutions modulo n . ...

Truncatable primes and unavoidable sets of divisors

Artūras Dubickas (2006)

Acta Mathematica Universitatis Ostraviensis

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We are interested whether there is a nonnegative integer u 0 and an infinite sequence of digits u 1 , u 2 , u 3 , ⋯ in base b such that the numbers u 0 b n + u 1 b n - 1 + ⋯ + u n - 1 b + u n , where n = 0 , 1 , 2 , ⋯ , are all prime or at least do not have prime divisors in a finite set of prime numbers S . If any such sequence contains infinitely many elements divisible by at least one prime number p ∈ S , then we call the set S unavoidable with respect to b . It was proved earlier that unavoidable sets in base b exist if b ∈ { 2 , 3 , 4 , 6 } , and that no unavoidable set exists in base b = 5 . Now,...

On sets which contain a qth power residue for almost all prime modules

Mariusz Ska/lba (2005)

Colloquium Mathematicae

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A classical theorem of M. Fried [2] asserts that if non-zero integers β ₁ , . . . , β l have the property that for each prime number p there exists a quadratic residue β j mod p then a certain product of an odd number of them is a square. We provide generalizations for power residues of degree n in two cases: 1) n is a prime, 2) n is a power of an odd prime. The proofs involve some combinatorial properties of finite Abelian groups and arithmetic results of [3].

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 ¬ ≡ 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 ∈ [ X i , X i + Y ] with X i θ i = Y , i = 1 , 2 , and θ 1 , θ 2 ≥ 0 . 933 such that ( p 1 + 2 ) ( p 2 + 2 ) has at most 9 prime factors.

On a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let p 1 , p 2 , ⋯ be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if k ≥ 5 , then ( p k + 1 - 1 ) ! ∣ ( 1 2 ( p k + 1 - 1 ) ) ! p k ! , which improves a previous result of the second author.

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 ≡ 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 .

Prime numbers along Rudin–Shapiro sequences

Christian Mauduit, Joël Rivat (2015)

Journal of the European Mathematical Society

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For a large class of digital functions f , we estimate the sums ∑ n ≤ x Λ ( n ) f ( n ) (and ∑ n ≤ x μ ( n ) f ( n ) , where Λ denotes the von Mangoldt function (and μ the Möbius function). We deduce from these estimates a Prime Number Theorem (and a Möbius randomness principle) for sequences of integers with digit properties including the Rudin-Shapiro sequence and some of its generalizations.

Aposyndesis in ℕ

José del Carmen Alberto-Domínguez, Gerardo Acosta, Maira Madriz-Mendoza (2023)

Commentationes Mathematicae Universitatis Carolinae

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We consider the Golomb and the Kirch topologies in the set of natural numbers. Among other results, we show that while with the Kirch topology every arithmetic progression is aposyndetic, in the Golomb topology only for those arithmetic progressions P ( a , b ) with the property that every prime number that divides a also divides b , it follows that being connected, being Brown, being totally Brown, and being aposyndetic are all equivalent. This characterizes the arithmetic progressions which are...

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 ∞ . 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 .