Divisor Sums of Generalised Exponential Polynomials

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.

Generalized Cullen numbers in linear 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.

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.

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.

One of the authors of the present paper, Kenji Nagasaka, considered, in his preceding article[4], various sampling procedures from the set of all positive integers and examined for the resulting sampled integers whether Benford's law holds or not. J. L. Brown, Jr. and R. L. Duncan [1] treated linear recurrence sequences and proved that, under several conditions on the corresponding characteristic equations, Benford's law is valid for certain linear recurrence integer sequences. It was shown by Lauwerens Kuipers and Jau-Shyong Shiue [3] that this result was able to be established by using one of J.G. van der Corput's difference theorems [5],[6]. Nagasaka succeeded in generalizing the main theorem of Duncan and Brown, which is Theorem 4.3 in [4]. Detailed study of linear recurrence sequences,, especially of order 2 is made in Theorem 4.1 and Theorem 4.2. But it still remains several cases ignored. In this joint paper, we shall adopt one of van der Corput's difference theorems as a main tool and prove some results on Benford's law for linear recurrence sequences. In the next Section, recurrence sequences of order 1 will be considered and we shall show sufficientconditions for Benford's law to be valid, which contain Theorem 3.2 in [4] as a special case. In Section 3 we shall give proofs of Theorem 4.1 and Theorem 4.2 in [4] based upon one of van der Corput's difference theorems. These Theorems do not contain the case where the corresponding characteristicequation has two complex conjugate roots. We shall show further that Benford's law holds for linear recurrence sequences when their corresponding characteristic equations have two purely imaginary conjugate roots. In the final Section, we shall consider general linear recurrence sequences of arbitrary order and prove analogous results as in the case of order 2.

The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.

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

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

Inspired by Faugere and Mou’s sparse FGLM algorithm, we show how using linear recurrent multi-dimensional sequences can allow one to perform operations such as the primary decomposition of an ideal, by computing of the annihilator of one or several such sequences.

Tropical recurrent sequences

The article discusses the standard course for learning recurrent sequences in the primary, secondary and high-school mathematics. We propose an extension of the curricula with adding recurrent linear homogeneous sequences in the upper high school course. Additionally, we present related problems in computer science and information technology in order to construct interdisciplinary links, aimed at demonstrating the applied side of recurrent sequences as well as motivating the more gifted students for the learning of some specific computer science basics.

We prove that for any base \(b\ge 2\) and for any linear homogeneous recurrence sequence \(\{a_n\}_{n\ge 1}\) satisfying certain conditions, there exits a positive constant \(c>0\) such that \(\# \{n\le x:\ a_n \;\text{ is} \text{ palindromic} \text{ in} \text{ base}\; b\} \ll x^{1-c}\).

With the advent of high-speed computing, there is a rekindled interest in the problem of determining when a given whole number N > 1 N > 1 is prime or composite. While complex algorithms have been developed to settle this for 200-digit numbers in a matter of minutes with a supercomputer, there is a need for simpler, more practical algorithms for dealing with numbers of a more modest size. Such practical tests for primality have recently been given (running in deterministic linear time) in terms of pseudoprimes for certain second- or third-order linear recurrence sequences. Here, a powerful general theory is described to characterize pseudoprimes for higher-order recurrence sequences. This characterization leads to a broadening and strengthening of practical primality tests based on such pseudoprimes.

Let $$(U_n)_{n\ge 0}$$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $$\ell $$ , let $$\ell \cdot 2^{\ell } + 1$$ be a Cullen number. Recently in Bilu et al. (J Number Theory 202:412–425, 2019), generalized Cullen numbers in terms of linear recurrence sequence $$(U_n)_{n\ge 0}$$ under certain weak assumptions has been studied. In this paper, we consider the more general Diophantine equation $$U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell } + Q(x)$$ , for a given polynomial $$Q(x) \in \mathbb {Z}[x]$$ and prove an effective finiteness result. Furthermore, we demonstrate our method by an example.