Divisors of the Gaussian integers in an arithmetic progression

Varbanec, P. D.; Zarzycki, Piotr

doi:10.1016/0022-314x(89)90003-6

Cited by 12 publications

(3 citation statements)

References 3 publications

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“…Then he showed that if S n is in AP, |S n | cannot be greater than 6 and finished the proof. For previous work on divisors in or not in AP, see [1,5] and on small divisors, see [2,4]. By Definition (1.1), 1 is in S n and this trivial divisor gives information about the AP.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

Chu¹

2020

Preprint

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Iannucci considered the positive divisors of a natural number n that do not exceed √ n and found all numbers whose such divisors are in arithmetic progression. In this paper, we generalize Iannucci's result by excluding the trivial divisors 1 and √ n from consideration. In other words, we loosen the condition Iannucci put on the arithmetic progression but are still able to characterize all natural numbers whose nontrivial, small divisors are in arithmetic progression. Surprisingly, though the condition is loosened, the length of our arithmetic progression cannot exceed 5.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

Chu¹

2020

Preprint

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“…Our work is a companion to a paper of Iannucci [3], who defined small divisors of n to be divisors not exceeding √ n and found all natural numbers whose small divisors are in arithmetic progression. For previous work on divisors in or not in arithmetic progression, see [1,6] and on small divisors, see [2,4].…”

Section: Introductionmentioning

confidence: 99%

When the Large Divisors of a Natural Number Are in Arithmetic Progression

Chu

2019

Preprint

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Iannucci considered the positive divisors of a natural number n that do not exceed the square root of n and found all numbers whose such divisors are in arithmetic progression. Continuing the work, we define large divisors to be divisors at least √ n and find all numbers whose large divisors are in arithmetic progression. The asymptotic formula for the count of these numbers up to a bound x is observed to be x log log x log x . 2010 Mathematics Subject Classification: 11B25

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“…In articles [5][6][7][8][9] the authors discussed special cases of sets of natural numbers A 1 , A 2 . The similar problem was considered over the ring of the Gaussian integers Z[i] in the work of Varbanets and Zarzycki [9] in case, when The following asymptotic formula was obtained where θ < 1 3 , α 1 is a number of form α 0 + βγ, β ∈ {0, ±1, ±i} with the smallest norm, the constant c(α 0 , γ) is computable and depends on α 0 and γ.…”

Section: Introductionmentioning

confidence: 99%

Divisor problem in special sets of Gaussian integers

Savastru¹

2016

Carpathian Math. Publ.

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DIVISOR PROBLEM IN SPECIAL SETS OF GAUSSIAN INTEGERSLet A 1 and A 2 be fixed sets of gaussian integers. We denote by τ A 1 ,A 2 (ω) the number of representations of ω in form ω = αβ, where α ∈ A 1 , β ∈ A 2 . We construct the asymptotical formula for summatory function τ A 1 ,A 2 (ω) in case, when ω lie in the arithmetic progression, A 1 is a fixed sector of complex plane, A 2 = Z[i].

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Divisors of the Gaussian integers in an arithmetic progression

Cited by 12 publications

References 3 publications

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

When the Nontrivial, Small Divisors of a Natural Number are in Arithmetic Progression

When the Large Divisors of a Natural Number Are in Arithmetic Progression

Divisor problem in special sets of Gaussian integers

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