Binary Recurrence Sequences Research Articles

Random balancing-like sequences

AbstractA balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

Bounds on the number of squares in recurrence sequences

We investigate the number of squares in a very broad family of binary recurrence sequences with u0=1. We show that there are at most two distinct squares in such sequences (the best possible result), except under very special conditions where we prove there are at most three such squares.

Near-squares in binary recurrence sequences

We call an integer a near-square if its absolute value is a square or a prime times a square. We investigate such near-squares in the binary recurrence sequences defined for integers [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. We show that for a given [Formula: see text], there is at most one [Formula: see text] such that [Formula: see text] is a near-square. With the exceptions of [Formula: see text] and [Formula: see text], any such [Formula: see text] can be a near-square only if [Formula: see text], [Formula: see text] is prime and [Formula: see text]. This is a part of a more general phenomenon regarding near-squares in nondegenerate recurrence sequences defined for the integers [Formula: see text] and [Formula: see text] by [Formula: see text], [Formula: see text] and [Formula: see text] for [Formula: see text]. This arises from a novel Aurifeuillean-type factorization of [Formula: see text] we have found.

The Solution of Fermat’s Two Squares Equation and Its Generalization In Lucas Sequences

As it is well known, there are an infinite number of primes in special forms such as Fermat's two squares form, p=x^2+y^2 or its generalization, p=x^2+y^4, where the unknowns x, y, and p represent integers. The main goal of this paper is to see if these forms still have an infinite number of solutions when the unknowns are derived from sequences with an infinite number of prime numbers in their terms. This paper focuses on the solutions to these forms where the unknowns represent terms in certain binary linear recurrence sequences known as the Lucas sequences of the first and second types.

On solutions of the Diophantine equation $L_n\pm L_m=p^a$

Lucas sequence is one of the most studied binary recurrence sequence defined by the relation $L_{n+2}=L_{n+1}+L_n;~L_0=2, L_1=1$. In this paper, we investigate all the sums and differences of two Lucas numbers that are powers of a odd prime $p$ satisfying $p<10^3$.

Additive Diophantine Equations with Binary Recurrences, {{mathcal {S}}}-Units and Several Factorials

There are many results in the literature concerning linear combinations of factorials among terms of linear recurrence sequences. Recently, Grossman and Luca provided effective bounds for such terms of binary recurrence sequences. In this paper we show that under certain conditions, even the greatest prime divisor of u_n-a_1m_1!-dots -a_km_k! tends to infinity, in an effective way. We give some applications of this result, as well.

On a variant of Pillai’s problem with binary recurrences and S-units

Let [Formula: see text] be a fixed binary recurrence with real characteristic roots [Formula: see text] satisfying [Formula: see text] and let [Formula: see text] be fixed distinct prime numbers. In this paper, we show that there exist effectively computable, positive constants [Formula: see text] and [Formula: see text] such that the Diophantine equation [Formula: see text] has at most [Formula: see text] solutions [Formula: see text] if [Formula: see text] and at most [Formula: see text] solutions [Formula: see text] if [Formula: see text]. In order to demonstrate the strength of our method we show that for the binary recurrence sequence [Formula: see text], with [Formula: see text] for [Formula: see text], [Formula: see text] and [Formula: see text], the Diophantine equation [Formula: see text] has at most six solutions [Formula: see text] unless [Formula: see text] in which case it has exactly seven solutions.

On prime factors of terms of binary recurrence sequences

We establish estimates from below for the greatest prime factor of the $n$th term of a non-degenerate binary recurrence sequence when the sequence belongs to a class of sequences which includes the Lucas sequences.

Diophantine triples with distinct binary recurrences

In this paper, we look at Diophantine triples with values in three different binary recurrence sequences. These are the Fibonacci and Pell sequences and the sequence of one more of powers of a given prime $p$. The novelty of the article is the appearance of three different sequences, as up to now the analogous problem had been investigated only for one sequence.

On the least common multiple of binary linear recurrence sequences

We present a method for estimating the least common multiple of a large class of binary linear recurrence sequences. Let P, Q, R0 and R1 be fixed integers, and let R=(Rn)n≥0 be the recurrence sequence defined by Rn+2=PRn+1−QRn for all n≥0. Under some conditions on the parameters, we determine a rational nontrivial divisor for Lk,n:= lcm(Rk,Rk+1,…,Rn), for all positive integers n and k, such that n≥k. As consequences, we derive nontrivial effective lower bounds for Lk,n, and we establish an asymptotic formula for log(Ln,n+m), where m is a fixed positive integer. Denoting by (Fn)n the usual Fibonacci sequence, we prove, for example, that for any m≥1, we have loglcm(Fn,Fn+1,…,Fn+m)∼n(m+1)logΦ( as n→+∞), where Φ denotes the golden ratio. We conclude the paper with some interesting identities and properties regarding the least common multiple of Lucas sequences.

Hyperelliptic curves and newform coefficients

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes ℓ<100 and all integers m≥1, we haveτ(n)≠±ℓ,±5m. For general newforms f with even integer weight 2k and integer coefficients, we prove for most integers j dividing 2k−1 and all ordinary primes p that af(p2) is never a j-th power. We prove a similar result for af(p4), conditional on the Frey-Mazur Conjecture. Our primary method involves relating questions about values of newforms to the existence of perfect powers in certain binary recurrence sequences, and makes use of bounds from the theory of linear forms in logarithms. The method extends without difficulty to a large family of Lebesgue-Nagell equations with fixed exponent. To prove results about general newforms, we also make use of the modular method and Ribet's level-lowering theorem.

Markov equation with components of some binary recurrent sequences

Markov equation with components of some binary recurrent sequences

On the l.c.m. of random terms of binary recurrence sequences

For every positive integer n and every δ∈[0,1], let B(n,δ) denote the probabilistic model in which a random set A⊆{1,…,n} is constructed by choosing independently every element of {1,…,n} with probability δ. Moreover, let (uk)k≥0 be an integer sequence satisfying uk=a1uk−1+a2uk−2, for every integer k≥2, where u0=0, u1≠0, and a1,a2 are fixed nonzero integers; and let α and β, with |α|≥|β|, be the two roots of the polynomial X2−a1X−a2. Also, assume that α/β is not a root of unity.We prove that, as δn/log⁡n→+∞, for every A in B(n,δ) we havelog⁡lcm(ua:a∈A)∼δLi2(1−δ)1−δ⋅3log⁡|α/(a12,a2)|π2⋅n2 with probability 1−o(1), where lcm denotes the lowest common multiple, Li2 is the dilogarithm, and the factor involving δ is meant to be equal to 1 when δ=1.This extends previous results of Akiyama, Tropak, Matiyasevich, Guy, Kiss and Mátyás, who studied the deterministic case δ=1, and is motivated by an asymptotic formula for lcm(A) due to Cilleruelo, Rué, Šarka, and Zumalacárregui.

Coincidence Between Two Binary Recurrent Sequences of Polynomials Arising from Diophantine Triples

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by 1 is a perfect square. A conjecture in this field asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple in some sense. This problem is reduced to study the coincidence between certain two binary recurrent sequences of integers. As an analogy of this, we consider a similar coincidence on the polynomial ring in one variable over the integers, and determine it completely. Our result is regarded as a generalization of a result in the paper “Complete solution of the polynomial version of a problem of Diophantus” by A. Dujella, C. Fuchs in Journal of Number Theory 106 (2004) on the polynomial variant of Diophantine tuples.

On special extrema of polynomials with applications to Diophantine problems

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial fin mathbb {Q}[x] can have at most one shift f(x)-lambda (lambda in mathbb {C}) of the form u(g(x))^q(h(x))^k with uin mathbb {C}, g,hin mathbb {C}[x] and either deg (g)=2, k is even, q=k/2 or deg (g)le 1, kge 2, qge 1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

Prime powers in sums of terms of binary recurrence sequences

Let $$(u_{n})_{n \ge 0}$$ be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we are interested in finding a finiteness result for the solutions of the Diophantine equation $$u_{n_{1}} + u_{n_{2}} + \cdots + u_{n_{t}} = p^{z}$$ with $$n_1> n_2> \cdots > n_t\ge 0$$ . Moreover, we explicitly find all the powers of three which are sums of three balancing numbers using lower bounds for linear forms in logarithms. Further, we use a variant of the Baker–Davenport reduction method in Diophantine approximation due to Dujella and Pethő.

Generalized Cullen numbers in linear recurrence sequences

A Cullen number is a number of the form m2m+1, where m is a positive integer. In 2004, Luca and Stănică proved, among other things, that the largest Fibonacci number in the Cullen sequence is F4=3. Actually, they searched for generalized Cullen numbers among some binary recurrence sequences. In this paper, we will work on higher order recurrence sequences. For a given linear recurrence (Gn)n, under weak assumptions, and a given polynomial T(x)∈Z[x], we shall prove that if Gn=mxm+T(x), then m≪1 and n≪log⁡|x|, where the implied constants depend only on (Gn)n and T(x).

Shifted powers in Lucas–Lehmer sequences

We develop a general framework for finding all perfect powers in sequences derived via shifting non-degenerate quadratic Lucas–Lehmer binary recurrence sequences by a fixed integer. By combining this setup with bounds for linear forms in logarithms and results based upon the modularity of elliptic curves defined over totally real fields, we are able to answer a question of Bugeaud, Luca, Mignotte and the third author by explicitly finding all perfect powers of the shape $$F_k \pm 2 $$ where $$F_k$$ is the k-th term in the Fibonacci sequence.

A lower bound for the two-variable Artin conjecture and prime divisors of recurrence sequences

In 1927, Artin conjectured that any integer other than −1 or a perfect square generates the multiplicative group (Z/pZ)× for infinitely many p. In 2000, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a proof by Hooley for the original conjecture. In this article, we prove an unconditional lower bound for this two-variable problem. In particular, we prove an estimate for the number of distinct primes which divide one of the first N terms of a non-degenerate binary recurrence sequence. We also prove a weaker version of the same theorem, and give three proofs that we consider to be of independent interest. The first proof uses a transcendence result of Stewart, the second uses a theorem of Bombieri and Schmidt on Thue equations and the third uses Mumford's gap principle for counting points on curves by their height. We finally prove a disjunction theorem, where we consider the set of primes satisfying either our two-variable condition or the original condition of Artin's conjecture. We give an unconditional lower bound for the number of such primes.

Diophantine triples in linear recurrences of Pisot type

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialised generalisations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.