On the diophantine equations
Carlo Viola (1973)
Acta Arithmetica
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Carlo Viola (1973)
Acta Arithmetica
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Florian Luca (2004)
Acta Arithmetica
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Susil Kumar Jena (2015)
Communications in Mathematics
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The two related Diophantine equations: and , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.
Florian Luca (2012)
Acta Arithmetica
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Mihai Cipu (2015)
Acta Arithmetica
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A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < ed < 1.55·1072b < 6.21·1035c = a + b + 2√(ab+1)...
Florian Luca, Volker Ziegler (2013)
Acta Arithmetica
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Given a binary recurrence , we consider the Diophantine equation with nonnegative integer unknowns , where for 1 ≤ i < j ≤ L, , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.
Andrzej Dąbrowski (2011)
Colloquium Mathematicae
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We completely solve the Diophantine equations (for q = 17, 29, 41). We also determine all and , where are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).
Min Tang, Quan-Hui Yang (2013)
Colloquium Mathematicae
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Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation has only the solution (x,y,z) = (1,1,1). We give an extension of this result.
Florian Luca, Alain Togbé (2009)
Colloquium Mathematicae
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We find all the solutions of the Diophantine equation in positive integers x,y,α,β,n ≥ 3 with x and y coprime.
Jerzy Browkin (2010)
Colloquium Mathematicae
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We consider systems of equations of the form and , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.
Attila Bérczes, Florian Luca, István Pink, Volker Ziegler (2016)
Acta Arithmetica
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Let g ≥ 2 be an integer and be the set of repdigits in base g. Let be the set of Diophantine triples with values in ; that is, is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set . We prove effective finiteness results for the set .
Jianping Wang, Tingting Wang, Wenpeng Zhang (2015)
Colloquium Mathematicae
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Let m be a positive integer. Using an upper bound for the solutions of generalized Ramanujan-Nagell equations given by Y. Bugeaud and T. N. Shorey, we prove that if 3 ∤ m, then the equation has only the positive integer solution (x,y,z) = (1,1,2).
Susil Kumar Jena (2018)
Communications in Mathematics
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In p. 219 of R.K. Guy’s , 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation has no integer solutions with and . But, contrary to this expectation, we show that for , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition .
Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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In this paper, we study triples and of distinct positive integers such that and are all three members of the same binary recurrence sequence.
Lingmin Liao, Michał Rams (2013)
Acta Arithmetica
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Let α be an irrational and φ: ℕ → ℝ⁺ be a function decreasing to zero. Let := y ∈ ℝ: ||nα -y|| < φ(n) for infinitely many n, where ||·|| denotes the distance to the nearest integer.
Zhongfeng Zhang, Jiagui Luo, Pingzhi Yuan (2012)
Colloquium Mathematicae
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Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either , k ≥ 1 or if c ≥ 2.