Diophantine equations concerning balancing and Lucas balancing numbers

Dey, Pallab Kanti; Rout, Sudhansu Sekhar

doi:10.1007/s00013-016-0994-z

Cited by 9 publications

(4 citation statements)

References 15 publications

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“…The case n 3 = 0. If n 2 = 0, then (2.3) becomes B n 1 = 3 z which is not true (see [10]). If n 2 > 0, then we have…”

Section: Proof Of Theoremmentioning

confidence: 99%

Prime powers in sums of terms of binary recurrence sequences

Mazumdar

Rout

2016

Preprint

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Let {u n } n≥0 be a non-degenerate binary recurrence sequence with positive discriminant and p be a fixed prime number. In this paper, we have shown a finiteness result for the solutions of the Diophantine equationMoreover, we explicitly find all the powers of three which are sums of three balancing numbers using the lower bounds for linear forms in logarithms. Further, we use a variant of Baker-Davenport reduction method in Diophantine approximation due to Dujella and Pethő.

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“…The case n 3 = 0. If n 2 = 0, then (2.3) becomes B n 1 = 3 z which is not true (see [10]). If n 2 > 0, then we have…”

Section: Proof Of Theoremmentioning

confidence: 99%

Prime powers in sums of terms of binary recurrence sequences

Mazumdar

Rout

2016

Preprint

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“…In [1], Ray solved some diophantine equations that involve balancing and Lucas balancing numbers. In [2], Dey and Rout found the perfect powers in the sequences of balancing and Lucas balancing numbers and identifed the Lucas balancing numbers which are products of a power of 3 and a perfect power. In addition, they proved that many diophantine equations that contains balancing and Lucas balancing numbers have no solutions.…”

Section: Introductionmentioning

confidence: 99%

Solutions of the Diophantine EquationsBr=Js+JtandCr
Gaber,
Ahmed
2023
Journal of Mathematics
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Let B r r ≥ 0 , J r r ≥ 0 , and C r r ≥ 0 be the balancing, Jacobsthal, and Lucas balancing numbers, respectively. In this paper, the diophantine equations B r = J s + J t and C r = J s + J t are completely solved. The solutions rely basically on Matveev’s theorem on linear forms in logarithms of algebraic numbers and a procedure of reducing the upper bound due to Dujella and Pethö.

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“…Recently, Bugeaud et al [7] proved that 0, 1, 8, and 144 are the only perfect powers in the Fibonacci sequence using linear forms in logarithm and modular approach. Similarly, perfect powers in balancing and Lucas balancing sequence has been studied (see [10]). Recently, the Diophantine equation…”

Section: Introductionmentioning

confidence: 99%