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On the arguments of the roots of the generalized Fibonacci polynomial
We revisit the classical subject of equidistribution of the roots of Littlewood-type polynomials. More precisely, we show that the roots of the...
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Root location for the characteristic polynomial of a Fibonacci type sequence
We analyse the roots of the polynomial x n − px n −1 − qx − 1 for p ≽ q ≽ 1. This is the characteristic polynomial of the recurrence relation F k,p,q ( n ) = pF...
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The Tribonacci Dirichlet series
We study some of the analytic properties of a Dirichlet series defined by the sequence of the Tribonacci numbers, such as its analytic continuation...
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On some combinatorial properties of generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions
We study generalized commutative Jacobsthal quaternions and generalized commutative Jacobsthal-Lucas quaternions. We present some properties of these...
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Sign changes of certain arithmetical function at prime powers
We examine an arithmetical function defined by recursion relations on the sequence { f ( p k )} k ∈ℕ and obtain sufficient condition(s) for the sequence to...
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On the Diophantine equation (2x − 1)(py − 1) = 2z2
Let p be an odd prime. By using the elementary methods we prove that: (1) if 2 ∤ x, p = ±3 (mod 8), the Diophantine equation (2 x − 1)( p y − 1) = 2 z 2 ...
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On the Third-Order Horadam and Geometric Mean Sequences
In this paper, in considering aspects of the geometric mean sequence, offers new results connecting generalized Tribonacci and third-order Horadam...
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Even perfect numbers in generalized Pell sequences
In this paper, by using linear forms in logarithms and the Baker–Davenport reduction procedure we prove that there are no even perfect numbers...
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Pell and Pell-Lucas Numbers of the Form −2a − 3b + 5c
In this paper, we find all Pell and Pell-Lucas numbers written in the form −2 a − 3 b + 5 c , in nonnegative integers a , b , c , with 0 ⩽ max{ a , b } ⩽ c .
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Lucas numbers as sums of two repdigits
In this paper, we determine all Lucas numbers that are sums of two repdigits. The largest one is L 14 = 843 = 66 + 777.
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Further results on Hilbert’s Tenth Problem
Hilbert’s Tenth Problem (HTP) asked for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has...
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Zeckendorf representations with at most two terms to x-coordinates of Pell equations
In this paper, we find all positive squarefree integers d satisfying that the Pell equation X 2 – dY 2 = ±1 has at least two positive integer solutions ( X ...
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Enumerating Some Stable Partitions Involving Stirling and r-Stirling Numbers of the Second Kind
The coefficient of the chromatic polynomial counts the number of partitions of the vertex set of a simple and finite graph G into k independent...
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Optimal codes from Fibonacci polynomials and secret sharing schemes
In this work, we study cyclic codes that have generators as Fibonacci polynomials over finite fields. We show that these cyclic codes in most cases...