On relative prime number in a sequence of positive integers

Ohtomo, Masahide; Tamari, Fumikazu

doi:10.1016/s0378-3758(02)00231-8

Cited by 7 publications

(13 citation statements)

References 5 publications

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“…For details and related results see [38] and [23]. In this paper we extend the investigations to recurrence sequences.…”

Section: Pillai Sequencesmentioning

confidence: 81%

“…As another direction of generalization, Ohtomo and Tamari [38] extended the problem of Pillai from consecutive integers to arithmetic progressions. For details and related results see [38] and [23].…”

Section: Pillai Sequencesmentioning

confidence: 99%

“…the papers of Caro [16], Saradha and Thangadurai [40] and Hajdu and Saradha [23] and the references there. In particular, Ohtomo and Tamari [38] have extended the original problem to arithmetic progressions, i.e. one considers k consecutive terms of an arithmetic progression, rather than k consecutive integers.…”

Section: Introductionmentioning

confidence: 99%

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On the GCD-s of k consecutive terms of Lucas sequences

Hajdu

Szikszai

2012

Journal of Number Theory

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Let u = (u n ) ∞ n=0 be a Lucas sequence, that is a binary linear recurrence sequence of integers with initial terms u 0 = 0 and u 1 = 1. We show that if k is large enough then one can find k consecutive terms of u such that none of them is relatively prime to all the others. We even give the exact values g u and G u for each u such that the above property first holds with k = g u ; and that it holds for all k G u , respectively. We prove similar results for Lehmer sequences as well, and also a generalization for linear recurrence divisibility sequences of arbitrarily large order. On our way to prove our main results, we provide a positive answer to a question of Beukers from 1980, concerning the sums of the multiplicities of 1 and −1 values in non-degenerate Lucas sequences. Our results yield an extension of a problem of Pillai from integers to recurrence sequences, as well.

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“…For details and related results see [38] and [23]. In this paper we extend the investigations to recurrence sequences.…”

Section: Pillai Sequencesmentioning

confidence: 81%

Section: Pillai Sequencesmentioning

confidence: 99%

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On the GCD-s of k consecutive terms of Lucas sequences

Hajdu

Szikszai

2012

Journal of Number Theory

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“…Evans [6] considered the case when s is an arithmetic progression and proved the existence of G s . Ohtomo and Tamari [12] derived the same, but also dealt with numerical aspects by showing that G s 384 for the sequence of odd integers. The most recent progress is due to Hajdu and Saradha [8] who gave an effective upper bound on G s depending only on the difference of the progression together with a heuristic algorithm to find the exact value of it, whenever the number of prime factors of the difference is 'small'.…”

Section: Introductionmentioning

confidence: 91%

A coprimality condition on consecutive values of polynomials

Sanna

Szikszai

2017

Bull. London Math. Soc.

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“…Due to Pillai's classical result from [22] and by a nice theorem of Brauer [3], we have that N is a Pillai sequence, with g N = G N = 17. Later, Ohtomo and Tamari [21] proved that for any coprime integers a, b the arithmetic progression an+b (n ≥ 1) is also a Pillai sequence. Recently, Hajdu and Szikszai [17] together with other related results proved that Lucas and Lehmer sequences of the first kind are Pillai sequences, as well.…”

Section: Introductionmentioning

confidence: 99%

On common factors within a series of consecutive terms of an elliptic divisibility sequence

Hajdu¹,

Szikszai²

2014

Publ. Math. Debrecen

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Abstract. We prove that for any elliptic divisibility sequence and any sufficiently large integer k, one can find k consecutive terms of the sequence such that none of these terms is coprime to all the others. In other words, elliptic divisibility sequences are Pillai sequences, named for a problem posed originally by Pillai for the sequence of integers. In fact we give an upper bound for the smallest value k0 past which this property is valid. We also provide a more general theorem where the coprimality condition is severely relaxed. In case of some particular sequences we give the values of k0, as well.

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On relative prime number in a sequence of positive integers

Cited by 7 publications

References 5 publications

On the GCD-s of k consecutive terms of Lucas sequences

On the GCD-s of k consecutive terms of Lucas sequences

A coprimality condition on consecutive values of polynomials

On common factors within a series of consecutive terms of an elliptic divisibility sequence

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