The Diophantine equation $(bn)^{x} + (2n)^{y} = ((b+2)n)^{z}$

Min Tang; Quan-Hui Yang

Displaying similar documents to “The Diophantine equation ${(b n)}^{x} + {(2 n)}^{y} = {((b + 2) n)}^{z}$ ”

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

Florian Luca, Alain Togbé (2009)

Colloquium Mathematicae

Similarity:

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.

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

Similarity:

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

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 < 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 $x^{2} = p^{a} \pm p^{b} + 1$

Florian Luca (2004)

Acta Arithmetica

Similarity:

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

A. Rotkiewicz, A. Schinzel (1987)

Colloquium Mathematicae

Similarity:

On the diophantine equations $\prod_{0}^{k} x_{i} - \sum_{0}^{k} x_{i} = n a n d \sum_{0}^{k} 1 / x_{i} = a / n$

Carlo Viola (1973)

Acta Arithmetica

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Sz. Tengely (2007)

Acta Arithmetica

Similarity:

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

Hui Lin Zhu (2011)

Acta Arithmetica

Similarity:

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

J. H. E. Cohn (2003)

Acta Arithmetica

Similarity:

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

Csaba Rakaczki (2012)

Acta Arithmetica

Similarity:

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

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

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

Similarity:

Given a binary recurrence ${u_{n}}_{n \geq 0}$ , we consider the Diophantine equation $u_{n_{1}}^{x_{1}} \dots u_{n_{L}}^{x_{L}} = 1$ with nonnegative integer unknowns $n_{1}, . . ., n_{L}$ , where $n_{i} \neq n_{j}$ for 1 ≤ i < j ≤ L, $m a x | x_{i} | : 1 \leq i \leq L \leq K$ , 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

Similarity:

We completely solve the Diophantine equations $x ² + 2^{a} q^{b} = y ⁿ$ (for q = 17, 29, 41). We also determine all $C = p ₁^{a ₁} \dots p_{k}^{a_{k}}$ and $C = 2^{a ₀} p ₁^{a ₁} \dots p_{k}^{a_{k}}$ , where $p ₁, . . ., p_{k}$ 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 terms $C x^{h} (h \geq 3)$ in Lucas sequences: an algorithm and applications to diophantine equations

Paulo Ribenboim (2003)

Acta Arithmetica

Similarity:

Displaying similar documents to “The Diophantine equation ( b n ) x + ( 2 n ) y = ( ( b + 2 ) n ) z ”

Displaying similar documents to “The Diophantine equation ${(b n)}^{x} + {(2 n)}^{y} = {((b + 2) n)}^{z}$ ”