On the Diophantine equation
A. Rotkiewicz, A. Schinzel (1987)
Colloquium Mathematicae
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A. Rotkiewicz, A. Schinzel (1987)
Colloquium Mathematicae
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Hui Lin Zhu (2011)
Acta Arithmetica
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Sz. Tengely (2007)
Acta Arithmetica
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Jiagui Luo (2001)
Acta Arithmetica
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J. H. E. Cohn (2003)
Acta Arithmetica
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Mihai Cipu, Tim Trudgian (2016)
Acta Arithmetica
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We consider Diophantine quintuples a, b, c, d, e. These are sets of positive integers, the product of any two elements of which is one less than a perfect square. It is conjectured that there are no Diophantine quintuples; we improve on current estimates to show that there are at most Diophantine quintuples.
Csaba Rakaczki (2012)
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.
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.
Luis V. Dieulefait (2005)
Acta Arithmetica
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Maciej Gawron (2013)
Colloquium Mathematicae
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We consider the Brocard-Ramanujan type Diophantine equation P(z) = n! + m!, where P is a polynomial with rational coefficients. We show that the ABC Conjecture implies that this equation has only finitely many integer solutions when d ≥ 2 and .
Peng Yang, Tianxin Cai (2012)
Acta Arithmetica
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Samir Siksek, John E. Cremona (2003)
Acta Arithmetica
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Maohua Le (2003)
Acta Arithmetica
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S. Akhtar Arif, Amal S. Al-Ali (2002)
Acta Arithmetica
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Susil Kumar Jena (2013)
Communications in Mathematics
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In this paper, the author shows a technique of generating an infinite number of coprime integral solutions for of the Diophantine equation for any positive integral values of when (mod 6) or (mod 6). For doing this, we will be using a published result of this author in The Mathematics Student, a periodical of the Indian Mathematical Society.
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.
Yongzhong Hu, Maohua Le (2015)
Acta Arithmetica
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Let a,b,c be fixed coprime positive integers with mina,b,c > 1, and let m = maxa,b,c. Using the Gel’fond-Baker method, we prove that all positive integer solutions (x,y,z) of the equation satisfy maxx,y,z < 155000(log m)³. Moreover, using that result, we prove that if a,b,c satisfy certain divisibility conditions and m is large enough, then the equation has at most one solution (x,y,z) with minx,y,z > 1.
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)...
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.