Salajan's conjecture on discriminating terms in an exponential sequence

Moree, Pieter; Zumalacárregui, Ana

doi:10.1016/j.jnt.2015.09.015

Cited by 10 publications

(30 citation statements)

References 15 publications

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“…that was first considered in this context by Jerzy Browkin. The case q = 5 was earlier dealt with by Moree and Zumalacárregui [9], who showed that, for this value of q, the smallest positive integer m discriminating u q (1), . .…”

Section: Putmentioning

confidence: 96%

On the discriminator of Lucas sequences

2018

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We consider the family of Lucas sequences uniquely determined by Un+2(k) = (4k + 2)Un+1(k) − Un(k), with initial values U0(k) = 0 and U1(k) = 1 and k ≥ 1 an arbitrary integer. For any integer n ≥ 1 the discriminator function D k (n) of Un(k) is defined as the smallest integer m such that U0(k), U1(k), . . . , Un−1(k) are pairwise incongruent modulo m. Numerical work of Shallit on D k (n) suggests that it has a relatively simple characterization. In this paper we will prove that this is indeed the case by showing that for every k ≥ 1 there is a constant n k such that D k (n) has a simple characterization for every n ≥ n k . The case k = 1 turns out to be fundamentally different from the case k > 1.

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Section: Putmentioning

confidence: 96%

On the discriminator of Lucas sequences

2018

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“…The value of q * does not play a role in comparing terms u q (i) and u q (j) with i and j of the same parity. For this reason, various results from [11] can be copied (almost) verbatim. In such a case we say that the proof follows by an equal parity index argument.…”

Section: Question 1 Does There Exist a Prime That Is Both Fermat Andmentioning

confidence: 99%

“…. , v(n) are pairwise incongruent modulo m. There are some results regarding the discriminator in case v(j) is a polynomial in j (see [11] and references therein). Beyond the polynomial case, very little is known.…”

Section: Introductionmentioning

confidence: 99%

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Browkin’s discriminator conjecture

Ciolan

Moree

2019

Colloq. Math.

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Let q ≥ 5 be a prime and put q * = (−1) (q−1)/2 ·q. We consider the integer sequence u q (1), u q (2), . . . , with u q (j) = (3 j − q * (−1) j )/4. No term in this sequence is repeated and thus for each n there is a smallest integer m such that u q (1), . . . , u q (n) are pairwise incongruent modulo m. We write D q (n) = m. The idea of considering the discriminator D q (n) is due to Browkin (2015) who, in case 3 is a primitive root modulo q, conjectured that the only values assumed by D q (n) are powers of 2 and of q. We show that this is true for n = 5, but false for infinitely many q in case n = 5. We also determine D q (n) in case 3 is not a primitive root modulo q.Browkin's inspiration for his conjecture came from earlier work of Moree and Zumalacárregui [11], who determined D 5 (n) for n ≥ 1, thus establishing a conjecture of Sȃlȃjan. For a fixed prime q their approach is easily generalized, but requires some innovations in order to deal with all primes q ≥ 7 and all n ≥ 1. Interestingly enough, Fermat and Mirimanoff primes play a special role in this.

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“…Recently, the discriminators of various types of functions have been considered by Zieve [10], Sun [8], Moree and Zumalacárregui [6], Haque and Shallit [4].…”

Section: Introductionmentioning

confidence: 99%

One some cancellation algorithms II

2017

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We define b f (n) to be the smallest integer (a natural number) d such that numbers f(n 1 , n 2 , ..., n m ), where n 1 +n 2 + ... + n m ≤ n are not divisible by d. For the given functions f : ℕ m → ℕ, we will obtain the asymptotic characterisation of the sequence of the least non canceled numbers (b f (n)) n∈ℕ . In the case f :∈ℕ, this characterisation can be rewritten in the terms of the permutations polynomials of finite commutative quotient ring ℤ/mℤ. There are situations in which we cannot expect formula for b f (n) to be simple, but we can provide the upper and lower bounds of it.

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Salajan's conjecture on discriminating terms in an exponential sequence

Abstract: Given a sequence of distinct positive integers v 1

Cited by 10 publications

References 15 publications

On the discriminator of Lucas sequences

On the discriminator of Lucas sequences

Browkin’s discriminator conjecture

One some cancellation algorithms II

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