This study investigates numbers that are powers of two and can be expressed as the sum of the squares of any two Pell numbers. We apply Baker's theory of linear forms in logarithms of algebraic numbers, along with...
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This study investigates numbers that are powers of two and can be expressed as the sum of the squares of any two Pell numbers. We apply Baker's theory of linear forms in logarithms of algebraic numbers, along with a variation of the Baker-Davenport reduction method, to solve the Diophantine equation P-m(2) + P-n(2) = 2a, where m,n, and a are non-negative integers, as presented here. Furthermore, the maple (c) codes used in the computations made throughout the paper are also shared.
Let (L-n((k)))(n >= 2-k) be the sequence of k-generalized Lucas numbers for some fixed integer k >= 2, whose first k terms are 0,& mldr;,0,2,1 and each term afterward is the sum of the preceding k terms. In ...
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Let (L-n((k)))(n >= 2-k) be the sequence of k-generalized Lucas numbers for some fixed integer k >= 2, whose first k terms are 0,& mldr;,0,2,1 and each term afterward is the sum of the preceding k terms. In this paper, we find all pairs of the k-generalized Lucas numbers that are multiplicatively dependent.
For k >= 2, the sequence (F-n((k)))(n >=-(k-2)) of k-generalized Fibonacci numbers is defined by the initial values 0, ..., 0, 1 = F-1((k)) and such that each term afterwards is the sum of the k preceding ones. ...
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For k >= 2, the sequence (F-n((k)))(n >=-(k-2)) of k-generalized Fibonacci numbers is defined by the initial values 0, ..., 0, 1 = F-1((k)) and such that each term afterwards is the sum of the k preceding ones. There are many recent results about the Diophantine equation (F-n((k)))(s) + (F-n+1((k)))(s) = F-m((l)), most of them dealing with the case k = l. In 2018, Bednarik et al. solved the equation for k <= l, but with s = 2. The aim of this paper is to continue this line of investigation by solving this equation for all s >= 2, but with (k, l) = (3, 2).
In this study, we show that if 2 < m < n and Fm Fn represents a repdigit, then (m, n) belongs to the set {(2, 2), (2, 3), (3, 3), (2, 4), (3, 4), (4, 4), (2, 5), (2, 6), (2, 10)}. Also, we show that if 0 < m ...
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In this study, we show that if 2 < m < n and Fm Fn represents a repdigit, then (m, n) belongs to the set {(2, 2), (2, 3), (3, 3), (2, 4), (3, 4), (4, 4), (2, 5), (2, 6), (2, 10)}. Also, we show that if 0 < m < n and Lm L represents a repdigit, then (m, n) belongs to the set (0, 0), (0, 1), (1, 1), (0, 2), (1, 2). (2, 2), (0, 3), (1, 3), (1, 4). (1, 5), (2,5), (3, 5), (4,5)}.
The k-generalized Pell sequence P-(k) := (P-n((k)))(n >= -(k-2)) is the linear recurrence sequence of order k, whose first k terms are 0, ..., 0, 1 and satisfies the relation P-n((k)) = 2P(n-1)((k)) + 2P(n-2)((k)) ...
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The k-generalized Pell sequence P-(k) := (P-n((k)))(n >= -(k-2)) is the linear recurrence sequence of order k, whose first k terms are 0, ..., 0, 1 and satisfies the relation P-n((k)) = 2P(n-1)((k)) + 2P(n-2)((k)) + ... + P-n-k((k)), for all n, k >= 2. In this paper, we investigate about integers that have at least two representations as a difference between a k-Pell number and a perfect power. In order to exhibit a solution method when b is known, we find all the integers c that have at least two representations of the form P-n((k)) - b(m) for b is an element of [2, 10]. This paper extends the previous works in Ddamulira et al. (Proc. Math. Sci. 127: 411-421, 2017) and Erazo et al. (J. Number Theory 203: 294-309, 2019).
Let (U (n) ) (n0) 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 ...
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Let (U (n) ) (n0) 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 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. Bertk, 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].
We find all positive integer solutions of the Diophantine equation F (n) + F (m) + F (e) = 2 (a) , where F (k) is the kth term of the Fibonacci sequence. This paper continues and extends the previous work of J.J. Brav...
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We find all positive integer solutions of the Diophantine equation F (n) + F (m) + F (e) = 2 (a) , where F (k) is the kth term of the Fibonacci sequence. This paper continues and extends the previous work of J.J. Bravo and F. Luca [On the Diophantine equation F (n) + F (m) = 2 (a) , Quaest. Math., to appear].
In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, (a), where F-k, is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bound...
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In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, (a), where F-k, is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.
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