A coprimality condition on consecutive values of polynomials

Sanna, Carlo; Szikszai, Márton

doi:10.1112/blms.12078

Cited by 2 publications

(2 citation statements)

References 13 publications

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“…Harrington and Jones [13] calculate g f for various families of quadratic polynomials, and conjecture that g f exists and is smaller than 35 for every quadratic polynomial f (they do not consider G f ). Sanna and Szikszai [19] prove the first part of this conjecture as a corollary of their main result that G f (and hence g f ) exists if f is quadratic or cubic. Moreover, they show that for every n ≥ G f there are infinitely many integers k ≥ 0 such that none of the consecutive terms f (k + 1), f (k + 2), .…”

Section: Introductionmentioning

confidence: 91%

“…Remark 2. Although our proof of Theorem 5 involves quadratic polynomials, Sanna and Szikszai's [19] theorem and proof that G f exists for all quadratic polynomials f ∈ Z[x] is not sufficient for our proof. It follows from their work and the ideas above that if L is a primitive Gaussian line, then there is an integer G L such that for all n ≥ G L there are infinitely integers k ≥ 0 with the property that none of the n rational integers N (α k+1 ), N (α k+2 ), .…”

Section: Gaussian Lines Are Pillai Sequencesmentioning

confidence: 99%

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Extending a problem of Pillai to Gaussian lines

Magness¹,

Nugent²,

Robertson³

2022

Preprint

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Let L be a primitive Gaussian line, that is, a line in the complex plane that contains two, and hence infinitely many, coprime Gaussian integers. We prove that there exists an integer G L such that for every integer n ≥ G L there are infinitely many sequences of n consecutive Gaussian integers on L with the property that none of the Gaussian integers in the sequence is coprime to all the others. We also investigate the smallest integer g L such that L contains a sequence of g L consecutive Gaussian integers with this property. We show that g L = G L in general. Also, g L ≥ 7 for every Gaussian line L, and we give necessary and sufficient conditions for g L = 7 and describe infinitely many Gaussian lines with g L ≥ 260, 000. We conjecture that both g L and G L can be arbitrarily large. Our results extend a well-known problem of Pillai from the rational integers to the Gaussian integers.

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Section: Introductionmentioning

confidence: 91%

Section: Gaussian Lines Are Pillai Sequencesmentioning

confidence: 99%

Extending a problem of Pillai to Gaussian lines

Magness¹,

Nugent²,

Robertson³

2022

Preprint

View full text Add to dashboard Cite

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Long gaps in sieved sets

et al. 2020

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For each prime p , let I_p \subset \mathbb{Z}/p\mathbb{Z} denote a collection of residue classes modulo p such that the cardinalities |I_p| are bounded and about 1 on average. We show that for sufficiently large x , the sifted set \{n \in \mathbb{Z}: n (\mathrm {mod}\: p) \not \in I_p \: \mathrm {for \: all} p \leq x\} contains gaps of size x (\log x)^{\delta} depends only on the densitiy of primes for which I_p \neq \emptyset . This improves on the "trivial'' bound of \gg x . As a consequence, for any non-constant polynomial f: \mathbb{Z} \to \mathbb{Z} with positive leading coefficient, the set \{n \leq X: f(n) \; \mathrm {composite}\} contains an interval of consecutive integers of length \ge (\log X) (\log\log X)^{\delta} for sufficiently large X , where \delta > 0 depends only on the degree of f .

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A coprimality condition on consecutive values of polynomials

Cited by 2 publications

References 13 publications

Extending a problem of Pillai to Gaussian lines

Extending a problem of Pillai to Gaussian lines

Long gaps in sieved sets

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