Displaying similar documents to “On the diophantine equations 0 k x i - 0 k x i = n a n d 0 k 1 / x i = a / n

On X 1 4 + 4 X 2 4 = X 3 8 + 4 X 4 8 and Y 1 4 = Y 2 4 + Y 3 4 + 4 Y 4 4

Susil Kumar Jena (2015)

Communications in Mathematics

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The two related Diophantine equations: X 1 4 + 4 X 2 4 = X 3 8 + 4 X 4 8 and Y 1 4 = Y 2 4 + Y 3 4 + 4 Y 4 4 , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.

Further remarks on Diophantine quintuples

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 < e s a t i s f i e s d < 1.55·1072 a n d b < 6.21·1035 w h e n 4 a < b , w h i l e f o r b < 4 a o n e h a s e i t h e r c = a + b + 2√(ab+1)...

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

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Given a binary recurrence u n n 0 , we consider the Diophantine equation u n 1 x 1 u n L x L = 1 with nonnegative integer unknowns n 1 , . . . , n L , where n i n j for 1 ≤ i < j ≤ L, m a x | x i | : 1 i L K , 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.

On the Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

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We completely solve the Diophantine equations x ² + 2 a q b = y (for q = 17, 29, 41). We also determine all C = p a p k a k and C = 2 a p a p k a k , where p , . . . , p k 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).

The Diophantine equation ( b n ) x + ( 2 n ) y = ( ( b + 2 ) n ) z

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 b x + 2 y = ( b + 2 ) z has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

On systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

Colloquium Mathematicae

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We consider systems of equations of the form x i + x j = x k and x i · x j = x k , 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.

Finiteness results for Diophantine triples with repdigit values

Attila Bérczes, Florian Luca, István Pink, Volker Ziegler (2016)

Acta Arithmetica

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Let g ≥ 2 be an integer and g be the set of repdigits in base g. Let g be the set of Diophantine triples with values in g ; that is, g 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 g . We prove effective finiteness results for the set g .

A note on the exponential Diophantine equation ( 4 m ² + 1 ) x + ( 5 m ² - 1 ) y = ( 3 m ) z

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 ( 4 m ² + 1 ) x + ( 5 m ² - 1 ) y = ( 3 m ) z has only the positive integer solution (x,y,z) = (1,1,2).

On the diophantine equation x y - y x = c z

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 x y - y x = c z with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either ( x , y , z ) = ( c k + 1 , 1 , k ) , k ≥ 1 or 2 x < y m a x 1 . 5 × 10 10 , c if c ≥ 2.

Inhomogeneous Diophantine approximation with general error functions

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 ω ( α ) : = s u p θ 1 : l i m i n f n n θ | | n α | = 0 . F o r a n y α w i t h a g i v e n ω ( α ) , w e g i v e s o m e s h a r p e s t i m a t e s f o r t h e H a u s d o r f f d i m e n s i o n o f t h e s e t E φ ( α ) := y ∈ ℝ: ||nα -y|| < φ(n) for infinitely many n, where ||·|| denotes the distance to the nearest integer.

Diophantine triples with values in binary recurrences

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 a , b and c of distinct positive integers such that a b + 1 , a c + 1 and b c + 1 are all three members of the same binary recurrence sequence.

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

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We study the Diophantine equations ( k ! ) n - k n = ( n ! ) k - n k and ( k ! ) n + k n = ( n ! ) k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or ( k , n ) = ( 1 , 2 ) , ( 2 , 1 ) and that the second one holds if and only if k = n .

The Diophantine Equation X³ = u+v over Real Quadratic Fields

Takaaki Kagawa (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let k be a real quadratic field and let k and k × be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ( X k , u , v k × ) is developed.

Diophantine equations and class number of imaginary quadratic fields

Zhenfu Cao, Xiaolei Dong (2000)

Discussiones Mathematicae - General Algebra and Applications

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Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and μ i - 1 , 1 ( i = 1 , 2 ) , and let h ( - 2 1 - e D ) ( e = 0 o r 1 ) denote the class number of the imaginary quadratic field ( ( - 2 1 - e D ) ) . 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 h ( - 2 1 - e D ) 0 ( m o d n ) , where D, and n satisfy k - 2 e + 1 = D x ² , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.

A note on the article by F. Luca “On the system of Diophantine equations a ² + b ² = ( m ² + 1 ) r and a x + b y = ( m ² + 1 ) z ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

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Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying A + B ( - 1 ) = ( m + ( - 1 ) ) r . We prove that the Diophantine equation | A | x + | B | y = ( m ² + 1 ) z has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever r > 10 74 or m > 10 34 . Our result is an explicit refinement of a theorem due to F. Luca.