Maximum gap in cyclotomic polynomials

Al-Kateeb, Ala'a; Ambrosino, Mary; Hong, Hoon; Lee, Eunjeong

doi:10.1016/j.jnt.2021.04.013

Cited by 4 publications

(9 citation statements)

References 31 publications

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“…• d is an integer such that d ≥ 1 8 n 1−ǫ/3 and d ≥ n 1−ǫ |J|; • for each a ∈ J \ {0}, the subgroup H generated by a has index [Z n : H] ≤ d − 1 24 n 1−ǫ/3 . Note that if we do not assume that J ∩ (−J) = {0}, Theorem 1.8 is simply implied by Proposition 2.10.…”

Section: |H|mentioning

confidence: 99%

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Intersective sets over abelian groups

Xu¹,

Yip²

2022

Preprint

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Given a finite abelian group G and a subset J ⊂ G with 0 ∈ J, let D G (J, N ) be the maximum size of A ⊂ G N such that the difference set A − A and J N have no non-trivial intersection. Recently, this extremal problem has been widely studied for different groups G and subsets J. In this paper, we generalize and improve the relevant results by Alon and by Hegedűs by building a bridge between this problem and cyclotomic polynomials. In particular, we construct infinitely many non-trivial families of G and J for which the upper bounds on D G (J, N ) obtained by them (via linear algebra method) can be improved exponentially. We also obtain a new upper bound D Fp ({0, 1}, N ) ≤ ( 1 2 + o( 1))(p − 1) N , which improves the previously best known result by Huang, Klurman and Pohoata. Our main tools are from algebra, number theory, and probability.

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Section: |H|mentioning

confidence: 99%

“…(4) Recently, Al-Kateeb, Ambrosino, Hong, and Lee [1] proved that the maximum gap between two consecutive exponents in Φ mp (t) is φ(m) for every prime number p and every square-free odd positive integer m less than p. Additionally, assume that φ(m) < m/2. Let n = pm, h(t) = Φ n (t) and J = supp(h).…”

mentioning

confidence: 99%

Intersective sets over abelian groups

Xu¹,

Yip²

2022

Preprint

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“…In 2016, Zhang [21] gave a simpler proof, along with the result on the number of occurrences of the maximum gaps. In 2021, Al-Kateeb et al [3] proved that g(Φ mp ) = φ(m), where m is a square-free odd integer and p > m is prime number.…”

Section: Introductionmentioning

confidence: 99%

“…Hence, a natural question is whether there is a formula for the number of nonzero terms of a ternary cyclotomic polynomials Φ pqr , where p < q < r are three odd prime numbers, and ultimately, arbitrary cyclotomic polynomials? In 2014, Bezdega [8] proved that the Hamming weight of the cyclotomic polynomial Φ n (x) is greater than or equal to n 1 3 . In 2016, A. Al-Kateeb [1] investigated the number of terms for a ternary cyclotomic polynomial and the following theorem was given (Theorem 7.1).…”

Section: Introductionmentioning

confidence: 99%

On the number of terms of some families of the ternary cyclotomic polynomials Φ3p2p3

Al-Kateeb,

Dagher

2023

Rad Hrvatske akademije znanosti i umjetnosti. Matematičke znano

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“…It is believed that the distribution of prime numbers must follow some rules, although it shows a certain randomness, that is, it has a certain degree of chaotic characteristics. Thus, the prime distribution problem has always been a confusing major topic, in which the difference between consecutive primes considered as an important characteristic of the distribution of the prime numbers attracted attentions of many researchers [16,17,18,19,20,21,22,23]. In theory, the interval between adjacent prime numbers can be arbitrarily large, however, people are more concerned about the following issues: 1) Let g be any even number, is there certainly two adjacent prime numbers whose interval is g?…”

Section: Introductionmentioning

confidence: 99%

Hierarchical Model-based Prediction on the Maximum Gap between Consecutive Primes below an Arbitrary Number

Liu

2022

ARJOM

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The distribution of the primes in natural number sequence has no simple law, which seems to show a certain degree of randomness. In this paper, we found a hierarchical progression pattern that the prime intervals are gradually tend to be evenly distributed on different orbits belongingto different levels, meanwhile, the prime numbers tend to give priority to spread all over each orbit of lower-energy level, and then gradually fill the orbits of higher-energy level with the expansion of prime number range. Moreover, by analyzing the count of prime numbers in different energy levels and different orbits based on the established hierarchical progression model, we investigated that the count of prime numbers in different energy levels decreases exponentially with the increase of energy levels, and its natural logarithm is approximately linear with the change of energy level. Based on these findings, we propose two kinds of strategies to estimate the value of maximum gap between consecutive primes lessen than a given number. Our developed results reveal that the prime distribution shows a layered increasing law, and give a reliable prediction guide to understand the upper limit of maximum interval below an arbitrary number.

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Maximum gap in cyclotomic polynomials

Cited by 4 publications

References 31 publications

Intersective sets over abelian groups

Intersective sets over abelian groups

On the number of terms of some families of the ternary cyclotomic polynomials Φ3p2p3

Hierarchical Model-based Prediction on the Maximum Gap between Consecutive Primes below an Arbitrary Number

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