Binary Recurrence Sequences Research Articles

Effective resolution of Diophantine equations of the form $$u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}$$ u n + u m = w p 1 z 1 ⋯ p s z s

Let $$u_n$$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, w a fixed non-zero integer and $$p_1,p_2,\ldots ,p_s$$ fixed, distinct prime numbers. In this paper we consider the Diophantine equation $$u_n+u_m=w p_1^{z_1} \ldots p_s^{z_s}$$ and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for n, m and $$z_1, \ldots , z_s$$ . Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.

Linear combinations of prime powers in sums of terms of binary recurrence sequences

Let (U n ) n≥0 be a nondegenerate binary recurrence sequence with positive discriminant. Let p 1 , . . . , p s be fixed prime numbers, b 1 , . . . , b s be fixed nonnegative integers, and a 1 , . . . , a t be positive integers. In this paper, under certain assumptions, we obtain a finiteness result for the solution of the Diophantine equation $$ {\alpha}_1{U}_{n1}+\cdots +{\alpha}_t{U}_{n1}={b}_1{p}_1^{z_1}+\cdots {b}_s{p}_s^{z_s}. $$ Moreover, we explicitly solve the equation F n1 + F n2 = 2 z1 + 3 z2 in nonnegative integers n 1, n 2, z 1, z 2 with z 2 ≥ z 1. The main tools used in this work are the lower bound for linear forms in logarithms and the Baker–Davenport reduction method. This work generalizes the recent papers [E. Mazumdar and S.S. Rout, Prime powers in sums of terms of binary recurrence sequences, arXiv:1610.02774] and [C. Bertok, L. Hajdu, I. Pink, and Z. Rabai, Linear combinations of prime powers in binary recurrence sequences, Int. J. Number Theory, 13(2):261–271, 2017].

Linear combinations of prime powers in binary recurrence sequences

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.

Multiplicative Diophantine equations with factors from different Lucas sequences

Let k≥2 and {un(1)}n≥0,…,{un(k)}n≥0 be k different nondegenerate binary recurrent sequences of integers. In this paper, we show that under certain conditions, there are only finitely many of k-tuples of the form (un1(1),…,unk(k)) whose contents are multiplicatively dependent. Furthermore, we determine all such instances when k=3 and the three sequences are the sequence of Fibonacci numbers, Pell numbers, and base 10 repunits, respectively.

Shifted powers in binary recurrence sequences

AbstractLet {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ ℤ) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form uk = xn+c which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over ℚ. We illustrate this approach by showing that the quaternary Diophantine equation x2n±6xn + 1 = 8y2 has no solutions in positive integers x, y, n with x, n > 1.

Linear combinations of factorials and $$S$$ S -units in a binary recurrence sequence

In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an \(S\)-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an \(S\)-unit associated to the set of primes \(\{2,3,5,7\}\).

On the sumset of binary recurrence sequences

On the sumset of binary recurrence sequences

D(4)-pair and its extension

Abstract We prove that if k ⩾ 3 , c and d are positive integers with c d and the set { k − 2 , k + 2 , c , d } has the property that the product of any of its distinct elements increased by 4 is a perfect square, then d is uniquely determined. In the proof we use the standard methods used in solving similar problems. Namely, we firstly transform our problem into solving the system of simultaneous pellian equations which furthermore leads to finding intersection of binary recurrence sequences. We get the lower bound for solutions using mostly the congruence method. Combining it with hypergeometric method in which we give an improvement of known results in our special case we get the desired result for “large” values of parameter k . The remaining values of k are solved using Bakerʼs theory of linear form in logarithms.

-FUNCTIONS OF ELLIPTIC CURVES AND BINARY RECURRENCES

AbstractLet $L(s, E)= {\mathop{\sum }\nolimits}_{n\geq 1} {a}_{n} {n}^{- s} $ be the $L$-series corresponding to an elliptic curve $E$ defined over $ \mathbb{Q} $ and $\mathbf{u} = \mathop{\{ {u}_{m} \} }\nolimits_{m\geq 0} $ be a nondegenerate binary recurrence sequence. We prove that if ${ \mathcal{M} }_{E} $ is the set of $n$ such that ${a}_{n} \not = 0$ and ${ \mathcal{N} }_{E} $ is the subset of $n\in { \mathcal{M} }_{E} $ such that $\vert {a}_{n} \vert = \vert {u}_{m} \vert $ holds with some integer $m\geq 0$, then ${ \mathcal{N} }_{E} $ is of density $0$ as a subset of ${ \mathcal{M} }_{E} $.

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

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.

A binary linear recurrence sequence of composite numbers

Let ( a , b ) ∈ Z 2 , where b ≠ 0 and ( a , b ) ≠ ( ± 2 , − 1 ) . We prove that then there exist two positive relatively prime composite integers x 1 , x 2 such that the sequence given by x n + 1 = a x n + b x n − 1 , n = 2 , 3 , … , consists of composite terms only, i.e., | x n | is a composite integer for each n ∈ N . In the proof of this result we use certain covering systems, divisibility sequences and, for some special pairs ( a , ± 1 ) , computer calculations. The paper is motivated by a result of Graham who proved this theorem in the special case of the Fibonacci-like sequence, where ( a , b ) = ( 1 , 1 ) .

Diophantine triples with values in binary recurrences

In this paper, we study triples a; b and c of distinct positive integers such that ab + 1; ac + 1 and bc + 1 are all three members of the same binary recurrence sequence.

Combinatorial numbers in binary recurrences

We give several effective and explicit results concerning the values of some polynomials in binary recurrence sequences. First we provide an effective finiteness theorem for certain combinatorial numbers (binomial coefficients, products of consecutive integers, power sums, alternating power sums) in binary recurrence sequences, under some assumptions. We also give an efficient algorithm (based on genus 1 curves) for determining the values of certain degree 4 polynomials in such sequences. Finally, partly by the help of this algorithm we completely determine all combinatorial numbers of the above type for the small values of the parameter involved in the Fibonacci, Lucas, Pell and associated Pell sequences.

ON PERFECT POWERS IN LUCAS SEQUENCES

Let (un)n≥0be the binary recurrence sequence of integers given by u0= 0, u1= 1 and un+2= 2(un+1+ un). We show that the only positive perfect powers in this sequence are u1= 1 and u4= 16. We further discuss the problem of determining perfect powers in Lucas sequences in general.

The Product of Like-Indexed Terms in Binary Recurrences

In recent work by Hajdu and Szalay, Diophantine equations of the form (ak−1)(bk−1)=x2 were completely solved for a few pairs (a, b). In this paper, a general finiteness theorem for equations of the form ukvk=xn is proved, where uk and vk are terms in certain types of binary recurrence sequences. Also, a unified computational approach for solving equations of the type (ak−1)(bk−1)=x2 is described, and this approach is used to completely solve such equations for almost all (a,b) in the range 1<a<b⩽100. In the final section of this paper, it is shown that the abc conjecture implies much stronger results on these types of Diophantine problems.

The Product of Like-Indexed Terms in Binary Recurrences

In recent work by Hajdu and Szalay, Diophantine equations of the form ( a k −1)( b k −1)= x 2 were completely solved for a few pairs ( a, b). In this paper, a general finiteness theorem for equations of the form u kv k = x n is proved, where u k and v k are terms in certain types of binary recurrence sequences. Also, a unified computational approach for solving equations of the type ( a k −1)( b k −1)= x 2 is described, and this approach is used to completely solve such equations for almost all ( a,b) in the range 1< a< b⩽100. In the final section of this paper, it is shown that the abc conjecture implies much stronger results on these types of Diophantine problems.

Sums of Factorials in Binary Recurrence Sequences

In this paper, we consider the problem of expressing a term of a given non-degenerate binary recurrence sequence as a sum of factorials. We show that if one bounds the number of factorials allowed, then there are only finitely many effectively computable terms which can be represented in this way. As an application, we also find the largest members of the classical Fibonacci and Lucas sequences which can be written as a sum or a difference of two factorials.

Divisibility properties of binary recurrent sequences

Divisibility properties of binary recurrent sequences

Products of Factorials in Binary Recurrence Sequences

Products of Factorials in Binary Recurrence Sequences

The multiplicity of binary recurrences

of the recurrence relation (0.1). It follows from the Skolem–Mahler–Lech Theorem that if U =∞ then one at least of ; and = is a root of unity. Consequently we call {um} nondegenerate if none of these quantities is a root of unity. In the thirties M. Ward conjectured that a nondegenerate binary recurrence sequence of integers um has multiplicity U 5 5. This conjecture was proved by Kubota [4]. Kubota also proved that any nondegenerate binary recurrence {um} with elements um in a number eld K of degree d has multiplicity U 5 c(d) (cf. [5]). Beukers and Tijdeman [2] established for this case the bound