The diophantine equation
Florian Luca (2004)
Acta Arithmetica
Similarity:
Florian Luca (2004)
Acta Arithmetica
Similarity:
H. L. Zhu (2012)
Acta Arithmetica
Similarity:
Susil Kumar Jena (2015)
Communications in Mathematics
Similarity:
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
Similarity:
Mihai Cipu (2015)
Acta Arithmetica
Similarity:
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
Similarity:
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
Similarity:
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
Similarity:
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
Similarity:
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
Similarity:
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
Similarity:
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
Similarity:
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).
Zhongfeng Zhang, Jiagui Luo, Pingzhi Yuan (2012)
Colloquium Mathematicae
Similarity:
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.
Lingmin Liao, Michał Rams (2013)
Acta Arithmetica
Similarity:
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.
Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
Similarity:
In this paper, we study triples and of distinct positive integers such that and are all three members of the same binary recurrence sequence.
Horst Alzer, Florian Luca (2017)
Mathematica Bohemica
Similarity:
We study the Diophantine equations and where and are positive integers. We show that the first one holds if and only if or and that the second one holds if and only if .
Takaaki Kagawa (2011)
Bulletin of the Polish Academy of Sciences. Mathematics
Similarity:
Let k be a real quadratic field and let and be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v (, ) is developed.
Zhenfu Cao, Xiaolei Dong (2000)
Discussiones Mathematicae - General Algebra and Applications
Similarity:
Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and , and let denote the class number of the imaginary quadratic field . In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then , where D, and n satisfy , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.
Takafumi Miyazaki (2014)
Acta Arithmetica
Similarity:
Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying . We prove that the Diophantine equation has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever or . Our result is an explicit refinement of a theorem due to F. Luca.