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Positive integer solutions of some Diophantine equations in terms of integer sequences
In this paper, we define some new number sequences, which we represent as $$ (B_{n}),(b_{n}),(y_{n})$$(Bn),(bn),(yn) and present relations of these new sequences with each other. Then, we give allExpand
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On the Diophantine equation x2 − kxy + y2 − 2n = 0
In this study, we determine when the Diophantine equation x2−kxy+y2−2n = 0 has an infinite number of positive integer solutions x and y for 0 ⩽ n ⩽ 10. Moreover, we give all positive integerExpand
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Some New Identities Concerning Generalized Fibonacciand Lucas Numbers
In this paper we obtain some identities containing generalized Fibonacciand Lucas numbers. Some of them are new and some are well known.By using some of these identities we give some congruencesExpand
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Repdigits base b as products of two Lucas numbers
Let (Ln ) be the sequence of Lucas numbers defined by L 0 = 2, L 1 = 1, and Ln = L n−1 + L n−2 for n ≥ 2. Let 0 ≤ m ≤ n and b = 2, 3, 4, 5, 6, 7, 8, 9. In this study, we show that if LmLn is a repd...
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Positive integer solutions of the diophantine equations $x^2 -5 F_n xy - 5(-1)^n y^2 = \pm 5^r$
In this study, we consider the Diophantine equations given in the title and determine when these equations have positive integer solutions. Moreover, we find all positive integer solutions of them inExpand
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An exponential Diophantine equation related to the difference of powers of two Fibonacci numbers.
In this paper, we prove that there is no x>=4 such that the difference of x-th powers of two consecutive Fibonacci numbers greater than 0 is a Lucas number.
On square classes in generalized Lucas sequences
Let P and Q be nonzero integers. Generalized Fibonacci and Lucas sequences are defined as follows: U0 = 0, U1 = 1, and Un+1 = PUn + QUn-1 for n ≥ 1 and V0 = 2, V1 = P, and Vn+1 = PVn + QVn-1 for n ≥Expand
Positive integer solutions of the diophantine equation x2 − Lnxy + (−1)ny2 = ±5r
In this paper, we consider the equation x2−Lnxy+(−1)ny2 = ±5r and determine the values of n for which the equation has positive integer solutions x and y. Moreover, we give all positive integerExpand
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