In this paper, we look at terms of Lucas sequences whose prime factors have indices with bounded gaps in the sequence of all prime numbers. Some of our results depend on certain widely believed conjectures. In our proofs we combine various tools, including Baker's method, the subspace theorem, and results of Stewart, and Murty and Wong. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).

Let E_1, … , E_s be s, not necessary distinct, elliptic curves over ℚ . We give upper bounds on the frequency of s-tuples of points in E_1(ℚ)×…× E_s(ℚ) whose denominators or x-coordinates are multiplicatively dependent. More precisely, we give such bounds in two scenarios: one in which we fix s non-torsion ℚ -rational points P_i ∈ E_i(ℚ) and arbitrary ℚ -rational points Q_i ∈ E_i(ℚ) , i =1, … , s , and we count s-tuples (n_1P_1+Q_1,… , n_sP_s+Q_s) ∈ E_1(ℚ) ×…× E_s(ℚ) with n_1, … , n_s in an arbitrary interval of length N, and the second in which we count points (P_1,… ,P_s) ∈ E_1(ℚ) ×…× E_s(ℚ) of bounded canonical height.

Let K be a number field, k≥ 2 an integer, (K^*)^k the k-fold direct product of K^* with coordinatewise multiplication, and Γ a finitely generated subgroup of rank r of (K^*)^k. Further, let H(α) denote the absolute exponential height of an algebraic number α. Fix non-zero elements a_1 a_k∈ K. We give asymptotic formulas for the number of 𝐱=(x_1 x_k) with H(a_1x_1+⋯ +a_kx_k)≤ X as X→∞ such that no non-empty subsum of a_1x_1+⋯ +a_kx_k vanishes. By the same method of proof, we obtain an asymptotic formula as X→∞ for the number of non-negative integers n with H(u_n)≤ X, where { u_n} is a linear recurrence sequence.

In this paper, we study the sets of integers which are n-th terms of Lucas sequences. We establish lower- and upper bounds for the size of these sets. These bounds are sharp for n sufficiently large. We also develop bounds on the growth order of the terms of Lucas sequences that are independent of the parameters of the sequence, which is a new feature.

Diophantine tuples are of ancient and modern interest, with a huge literature. In this paper, we study Diophantine graphs, that is, finite graphs whose vertices are distinct positive integers, and two vertices are linked by an edge if and only if their product increased by one is a square. We provide various results for Diophantine graphs, including extendability properties, lower‐ and upper bounds for the maximum number of edges and chromatic numbers.

Let $E_1, \ldots, E_s $ be $s$, not necessary distinct, elliptic curves over $\mathbb Q$. Given $s$ non-torsion $\mathbb Q$-rational points $P_i \in E_i(\mathbb Q)$ and arbitrary $\mathbb Q$-rational points $Q_i \in E_i(\mathbb Q)$, $i =1, \ldots, s$, we give an upper bound on the frequency of $s$-tuples \[ (n_1P_1+Q_1,\ldots, n_sP_s+Q_s) \in E_1(\mathbb Q) \times \ldots \times E_s(\mathbb Q) \] with $n_1, \ldots, n_s$ in an arbitrary interval of length $N$, whose denominators or $x$-coordinates are multiplicatively dependent.

For a wide class of integer linear recurrence sequences $(u(n))_{n=1}<^>\infty $ , we give an upper bound on the number of s-tuples $\left (n_1, \ldots , n_s\right ) \in \left ({\mathbb Z}\cap [M+1,M+ N]\right )<^>s$ such that the corresponding elements $u(n_1), \ldots , u(n_s)$ in the sequence are multiplicatively dependent.

This paper is the continuation of Hajdu and Tijdeman (Ramanujan J 66:Article 74, 2025), where we deal with Lucas sequences. Here we study integers represented by integer sequences which satisfy binary recursive relations. In the case of non-degenerate sequences we give an upper bound on the largest index of a zero term and bounds on the growth order of the absolute values of the terms, both only in terms of the two initial values, which is a novel feature. Some of these bounds are best possible apart from a multiplicative constant.

Let C_n be the n-th Catalan number. In this note, we prove that the product of two different Catalan numbers cannot be a square of an integer. On the other hand, for each k≥ 3 , there are infinitely many k-tuples of pairwise different Catalan numbers with product being squares. We also obtain a characterization of x∈ℕ_+ such that C_xC_x+1 is a power-full number and prove that there are infinitely many such x. Moreover we present some numerical results which motivate further problems.

In this paper, we find all solutions of the Diophantine equation F_n^x+F_k^x=F_m^y , where {F_m}_m≥ 0 is the Fibonacci sequence.

We study the square values of Littlewood polynomials. Using various methods we give all these values for the degrees n=3, 5 and n≤ 24 even. Beside this, we gather computational data (by providing all solutions in a certain range) for n odd with n≤ 17 . We propose some striking problems for further research, as well.

We find all solutions of three exponential Diophantine equations, arising from certain quadratic, cubic and quartic identities. The first identity comes from a painting of the famous Russian painter Nikolay Bogdanov-Belsky, highlighted by Ja. I. Perelman. The equations have five, four and six terms, respectively, so they cannot be handled by classical tools based upon Baker's method. To solve the equations we use our method developed earlier, which is based upon Skolem's conjecture, local considerations and a computational approach.

The structure as well as several arithmetic properties of the solution sets of norm form equations are of classical and recent interest. In this paper, we give a finiteness result for terms of linear recurrence sequences appearing in the coordinates of solutions of norm form equations. Our main theorem yields a common generalization of certain recent results from the literature.

In this paper we consider Diophantine equations of the form $f(x)=g(y)$ where $f$ has simple rational roots and $g$ has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when $g$ has only simple rational roots as well, we can give a simpler statement. Also we provide an application to equal products with terms belonging to blocks of consecutive integers of bounded length. The latter theorem is related to problems and results of Erdős and Turk, and of Erdős and Graham.

There are many results in the literature concerning polyno-mial values and (shifted) power values of polynomials with consecutive integer roots, or more generally, with roots form-ing an arithmetic progression. It is an interesting question that how far one can 'disturb' the structure of the roots such that the finiteness results still remain valid. Also there are many results into this direction, with adding or removing one or more terms (roots). In this paper we study a case where (part of) the symmet-ric root structure is preserved, however, we allow (possibly large) increasing gaps between the roots. We prove that the finiteness of the solutions can also be guaranteed under these generalized circumstances. In our proofs we combine Bak-er's method and the Bilu-Tichy theorem with a new result providing an increasing property of the extremal values of polynomials with distinct real roots satisfying certain sym-metry and increasing gap properties. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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!-… -a_km_k! tends to infinity, in an effective way. We give some applications of this result, as well.

We give finiteness results for the shifted power values and polynomial values of Littlewood polynomials.

In this paper, we study upper bounds for the degrees of polynomials with only rational roots. First, we assume that the coefficients are bounded. In the second theorem, we suppose that the primes 2 and 3 do not divide any coefficient. The third theorem concerns the case that all coefficients are composed of primes from a fixed finite set.

In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A \subseteq \{1, \dots , n\}$$ A ⊆ { 1 , ⋯ , n } in k-sets such that the first $$k-1$$ k - 1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial $$f_A(x) := \prod _{x \in A} (x-a)$$ f A ( x ) : = ∏ x ∈ A ( x - a ) . Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation $$f_A(x)=P(y)$$ f A ( x ) = P ( y ) where $$P(y)\in \mathbb {Q}[y]$$ P ( y ) ∈ Q [ y ] and on shifted power values of $$f_A(x)$$ f A ( x ) .

We give effective finiteness results for the power values of polynomials with coefficients composed of a fixed finite set of primes; in particular, of Littlewood polynomials.