Displaying similar documents to “Associated primes, integral closures and ideal topologies”

Consecutive primes in tuples

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

Acta Arithmetica

Similarity:

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...

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

Karin Halupczok (2013)

Journal de Théorie des Nombres de Bordeaux

Similarity:

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.

Generalization of the S -Noetherian concept

Abdelamir Dabbabi, Ali Benhissi (2023)

Archivum Mathematicum

Similarity:

Let A be a commutative ring and 𝒮 a multiplicative system of ideals. We say that A is 𝒮 -Noetherian, if for each ideal Q of A , there exist I 𝒮 and a finitely generated ideal F Q such that I Q F . In this paper, we study the transfer of this property to the polynomial ring and Nagata’s idealization.

On a system of equations with primes

Paolo Leonetti, Salvatore Tringali (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Given an integer n 3 , let u 1 , ... , u n be pairwise coprime integers 2 , 𝒟 a family of nonempty proper subsets of { 1 , ... , n } with “enough” elements, and ε a function 𝒟 { ± 1 } . Does there exist at least one prime q such that q divides i I u i - ε ( I ) for some I 𝒟 , but it does not divide u 1 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 { ± 1 } and A is a set of three or more primes that contains all prime divisors of any...

Towards Bauer's theorem for linear recurrence sequences

Mariusz Skałba (2003)

Colloquium Mathematicae

Similarity:

Consider a recurrence sequence ( x k ) k of integers satisfying x k + n = a n - 1 x k + n - 1 + . . . + a x k + 1 + a x k , where a , a , . . . , a n - 1 are fixed and a₀ ∈ -1,1. Assume that x k > 0 for all sufficiently large k. If there exists k₀∈ ℤ such that x k < 0 then for each negative integer -D there exist infinitely many rational primes q such that q | x k for some k ∈ ℕ and (-D/q) = -1.

A diophantine equation involving special prime numbers

Stoyan Dimitrov (2023)

Czechoslovak Mathematical Journal

Similarity:

Let [ · ] be the floor function. In this paper, we prove by asymptotic formula that when 1 < c < 3441 2539 , then every sufficiently large positive integer N can be represented in the form N = [ p 1 c ] + [ p 2 c ] + [ p 3 c ] + [ p 4 c ] + [ p 5 c ] , where p 1 , p 2 , p 3 , p 4 , p 5 are primes such that p 1 = x 2 + y 2 + 1 .

Elements of large order on varieties over prime finite fields

Mei-Chu Chang, Bryce Kerr, Igor E. Shparlinski, Umberto Zannier (2014)

Journal de Théorie des Nombres de Bordeaux

Similarity:

Let 𝒱 be a fixed algebraic variety defined by m polynomials in n variables with integer coefficients. We show that there exists a constant C ( 𝒱 ) such that for almost all primes p for all but at most C ( 𝒱 ) points on the reduction of 𝒱 modulo p at least one of the components has a large multiplicative order. This generalises several previous results and is a step towards a conjecture of B. Poonen.

On the least almost-prime in arithmetic progressions

Liuying Wu (2024)

Czechoslovak Mathematical Journal

Similarity:

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 .

Connectedness of some rings of quotients of C ( X ) with the m -topology

F. Azarpanah, M. Paimann, A. R. Salehi (2015)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In this article we define the m -topology on some rings of quotients of C ( X ) . Using this, we equip the classical ring of quotients q ( X ) of C ( X ) with the m -topology and we show that C ( X ) with the r -topology is in fact a subspace of q ( X ) with the m -topology. Characterization of the components of rings of quotients of C ( X ) is given and using this, it turns out that q ( X ) with the m -topology is connected if and only if X is a pseudocompact almost P -space, if and only if C ( X ) with r -topology is connected. We also...

Matrix coefficients, counting and primes for orbits of geometrically finite groups

Amir Mohammadi, Hee Oh (2015)

Journal of the European Mathematical Society

Similarity:

Let G : = SO ( n , 1 ) and Γ ( n - 1 ) / 2 for n = 2 , 3 and when δ > n - 2 for n 4 , we obtain an effective archimedean counting result for a discrete orbit of Γ in a homogeneous space H G where H is the trivial group, a symmetric subgroup or a horospherical subgroup. More precisely, we show that for any effectively well-rounded family { T H G } of compact subsets, there exists η > 0 such that # [ e ] Γ T = ( T ) + O ( ( T ) 1 - η ) for an explicit measure on H G which depends on Γ . We also apply the affine sieve and describe the distribution of almost primes on orbits of Γ in arithmetic...