A ternary Diophantine inequality over primes

Roger Baker; Andreas Weingartner

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 $| λ ₁ p ₁ + λ ₂ p ² ₂ + λ ₃ p ₃^{k} + ϖ | \leq {(m a x_{j} p_{j})}^{- (33 - 29 k) / (72 k) + ε}$ has infinitely many solutions in prime variables p₁, p₂, p₃.

A note on the Diophantine equation $(a^{n} x^{m} \pm 1) / (a^{n} x \pm 1) = y^{n} + 1$

Jiagui Luo (2001)

Acta Arithmetica

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On the Diophantine equation $x^{2} + q^{2 m} = 2 y^{p}$

Sz. Tengely (2007)

Acta Arithmetica

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A note on the Diophantine equation $x^{2} + q^{m} = y^{3}$

Hui Lin Zhu (2011)

Acta Arithmetica

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On the Diophantine equation $x^{2} + 7 = y^{m}$

Samir Siksek, John E. Cremona (2003)

Acta Arithmetica

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The diophantine equation $x^{2} + C = y^{n}$ , II

J. H. E. Cohn (2003)

Acta Arithmetica

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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.

On the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$

Florian Luca, Alain Togbé (2009)

Colloquium Mathematicae

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We find all the solutions of the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$ in positive integers x,y,α,β,n ≥ 3 with x and y coprime.

A conjecture concerning the exponential diophantine equation $a^{x} + b^{y} = c^{z}$

Maohua Le (2003)

Acta Arithmetica

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On the Diophantine equation $x^{p} + y^{2 p} = z^{2}$

A. Rotkiewicz, A. Schinzel (1987)

Colloquium Mathematicae

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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 some generalizations of the diophantine equation $s (1^{k} + 2^{k} + \dots + x^{k}) + r = d y^{n}$

Csaba Rakaczki (2012)

Acta Arithmetica

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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.

On the Diophantine equation $X^{2} - (2^{2 m} + 1) Y^{4} = - 2^{2 m}$

Michael Stoll, P. G. Walsh, Pingzhi Yuan (2009)

Acta Arithmetica

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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 $D x ² + 2^{2 m + 1} = y ⁿ$

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.

On the system of Diophantine equations $a^{2} + b^{2} = {(m^{2} + 1)}^{r}$ and $a^{x} + b^{y} = {(m^{2} + 1)}^{z}$

Florian Luca (2012)

Acta Arithmetica

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Corrigendum to the paper “A note on the Diophantine equation $x^{2} + q^{m} = y^{3}$ ” (Acta Arith. 146 (2011), 195-202)

H. L. Zhu (2012)

Acta Arithmetica

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On the diophantine equation $a x ² + b^{m} = 4 y ⁿ$

S. Akhtar Arif, Amal S. Al-Ali (2002)

Acta Arithmetica

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The diophantine equation $x^{2} = p^{a} \pm p^{b} + 1$

Florian Luca (2004)

Acta Arithmetica

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