On the Diophantine equation
Sz. Tengely (2007)
Acta Arithmetica
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Sz. Tengely (2007)
Acta Arithmetica
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Jiagui Luo (2001)
Acta Arithmetica
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Hui Lin Zhu (2011)
Acta Arithmetica
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Maohua Le (2003)
Acta Arithmetica
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J. H. E. Cohn (2003)
Acta Arithmetica
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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.
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).
Samir Siksek, John E. Cremona (2003)
Acta Arithmetica
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A. Rotkiewicz, A. Schinzel (1987)
Colloquium Mathematicae
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S. Akhtar Arif, Amal S. Al-Ali (2002)
Acta Arithmetica
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Roger Baker, Andreas Weingartner (2014)
Acta Arithmetica
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Let 1 < c < 10/9. For large real numbers R > 0, and a small constant η > 0, the inequality holds for many prime triples. This improves work of Kumchev [Acta Arith. 89 (1999)].
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.
Michael Stoll, P. G. Walsh, Pingzhi Yuan (2009)
Acta Arithmetica
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Florian Luca (2012)
Acta Arithmetica
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J. H. E. Cohn (2003)
Colloquium Mathematicae
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It is shown that for a given squarefree positive integer D, the equation of the title has no solutions in integers x > 0, m > 0, n ≥ 3 and y odd, nor unless D ≡ 14 (mod 16) in integers x > 0, m = 0, n ≥ 3, y > 0, provided in each case that n does not divide the class number of the imaginary quadratic field containing √(-2D), except for a small number of (stated) exceptions.
Csaba Rakaczki (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)...
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.
Luis V. Dieulefait (2005)
Acta Arithmetica
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Florian Luca (2004)
Acta Arithmetica
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N. Saradha, T. N. Shorey (2007)
Acta Arithmetica
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