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Displaying similar documents to “A ternary Diophantine inequality over primes”

On a ternary Diophantine problem with mixed powers of primes

Alessandro Languasco, Alessandro Zaccagnini (2013)

Acta Arithmetica

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Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality | λ p + λ p ² + λ p k + ϖ | ( m a x j p j ) - ( 33 - 29 k ) / ( 72 k ) + ε has infinitely many solutions in prime variables p₁, p₂, p₃.

A note on ternary purely exponential diophantine 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 a x + b y = c z 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.

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.

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 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)...

The Diophantine equation D x ² + 2 2 m + 1 = y

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.