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Let (P n) n ≥ 0 and (Q n) n ≥ 0 be the Pell and Pell–Lucas sequences respectively. Let b be a positive integer such that b ≥ 2 . In this paper, we investigate the solutions of the following two Diophantine equations: P n = b d Q m + Q k and Q n = b d P m + P k , where n, m, k and d are non-negative integers with a condition on d. More precisely, we prove that the above equations have only finitely many solutions in integers (m, n, k, b, d). We explicitly determine these solutions for the cases 2 ≤ b ≤ 10 as an application. [ABSTRACT FROM AUTHOR]

A b-repdigit is a positive integer that has only one distinct digit in its base b expansion, i.e. a number of the form a(bm – 1)=(b – 1), for some positive integers m ≥ 1, b ≥ 2 and 1 ≤ a ≤ b – 1. Let r; s be non-zero integers with r ≥ 1 and s ∈ {±1}, let {Un}n ≥0 be the Lucas sequence given by Un+2 = rUn+1 + sUn, with U0 = 0 and U1 = 1: In this paper, we give effective bounds for the solutions of the Diophantine equation where a; b; n; k and m are positive integers such that 1 ≤ a ≤ b – 1; n; k ≥ 1 and 2 ≤ b ≤ 10. Then, we explicitly solve the above Diophantine equation for the Fibonacci, Pell and balancing sequences. [ABSTRACT FROM AUTHOR]

Let k ≥ 2. A generalization of the well-known Pell sequence is the k-Pell sequence whose first k terms are 0,..., 0, 1 and each term afterwards is given by the linear recurrence p n (k) = 2 P n − 1 (k) + P n − 2 (k) + ⋯ + P n − k (k) . The goal of this paper is to show that 11, 33, 55, 88 and 99 are all repdigits expressible as sum or difference of two k-Pell. The proof of our main theorem uses lower bounds for linear forms in logarithms of algebraic numbers and a modified version of Baker-Davenport reduction method (due to Dujella and Pethő). This extends a result of Bravo and Herrera [Repdigits in generalized Pell sequences, Arch. Math. (Brno) 56(4) (2020), 249–262]. [ABSTRACT FROM AUTHOR]

Let Pk and Qk be the k-th terms of the Pell and Pell-Lucas sequences respectively. In this paper, we find all the solutions of the Diophantine equations P n + 1 x ± P n − 1 x = Q m in nonnegative integer variables (m, n, x). [ABSTRACT FROM AUTHOR]

In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence. [ABSTRACT FROM AUTHOR]

In this paper, we determine all Lucas numbers that are sums of two repdigits. The largest one is L14 = 843 = 66 + 777. [ABSTRACT FROM AUTHOR]

In this paper, we consider the $$D(\pm k)$$ -triple $$\{k\mp 1,k, 4k\mp 1\}$$ and we prove that, if k is not a perfect square then: This extends a work done by Fujita [13]. [ABSTRACT FROM AUTHOR]

In this paper, we find all the solutions of the title Diophantine equation in positive integers (m, n, k, x), where Pi is the ith term of the Pell sequence. [ABSTRACT FROM AUTHOR]

Dujella and Pethő, generalizing a result of Baker and Davenport, proved that the set { 1 , 3 } cannot be extended to a Diophantine quintuple. As a consequence of our main result, we show that the Diophantine pair { 1 , b } is regular if b − 1 is a prime power. [ABSTRACT FROM AUTHOR]