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

Zhongfeng Zhang; Jiagui Luo; Pingzhi Yuan

Displaying similar documents to “On the diophantine equation $x^{y} - y^{x} = c^{z}$ ”

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 $k = n$ .

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 $_{g} \subset ℕ$ 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 $_{g}$ .

On some Diophantine equations involving balancing numbers

Euloge Tchammou, Alain Togbé (2021)

Archivum Mathematicum

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In this paper, we find all the solutions of the Diophantine equation $B_{1}^{p} + 2 B_{2}^{p} + \dots + k B_{k}^{p} = B_{n}^{q}$ in positive integer variables $(k, n)$ , where $B_{i}$ is the $i^{t h}$ balancing number if the exponents $p$ , $q$ are included in the set ${1, 2}$ .

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 $A + B \sqrt (- 1) = {(m + \sqrt (- 1))}^{r}$ . 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.

A remark on a Diophantine equation of S. S. Pillai

Azizul Hoque (2024)

Czechoslovak Mathematical Journal

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S. S. Pillai proved that for a fixed positive integer $a$ , the exponential Diophantine equation $x^{y} - y^{x} = a$ , $min (x, y) > 1$ , has only finitely many solutions in integers $x$ and $y$ . We prove that when $a$ is of the form $2 z^{2}$ , the above equation has no solution in integers $x$ and $y$ with $gcd (x, y) = 1$ .

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 $x_{i} \cdot x_{j} = x_{k}$ , 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.

Complete solution of the Diophantine equation $x^{y} + y^{x} = z^{z}$

Mihai Cipu (2019)

Czechoslovak Mathematical Journal

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The triples $(x, y, z) = (1, z^{z} - 1, z)$ , $(x, y, z) = (z^{z} - 1, 1, z)$ , where $z \in ℕ$ , satisfy the equation $x^{y} + y^{x} = z^{z}$ . In this paper it is shown that the same equation has no integer solution with $min {x, y, z} > 1$ , thus a conjecture put forward by Z. Zhang, J. Luo, P. Z. Yuan (2013) is confirmed.

On the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$

Ruizhou Tong (2021)

Czechoslovak Mathematical Journal

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Let $p$ be an odd prime. By using the elementary methods we prove that: (1) if $2 ∤ x$ , $p \equiv \pm 3 (mod 8),$ the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$ has no positive integer solution except when $p = 3$ or $p$ is of the form $p = 2 a_{0}^{2} + 1$ , where $a_{0} > 1$ is an odd positive integer. (2) if $2 ∤ x$ , $2 ∣ y$ , $y \neq 2, 4,$ then the Diophantine equation $(2^{x} - 1) (p^{y} - 1) = 2 z^{2}$ has no positive integer solution.

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.

On $x^{n} + y^{n} = n! z^{n}$

Susil Kumar Jena (2018)

Communications in Mathematics

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In p. 219 of R.K. Guy’s , 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^{n} + y^{n} = n! z^{n}$ has no integer solutions with $n \in ℕ_{+}$ and $n > 2$ . But, contrary to this expectation, we show that for $n = 3$ , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition $gcd (x, y, z) = 1$ .

On the Diophantine equation $\sum_{j = 1}^{k} j F_{j}^{p} = F_{n}^{q}$

Gökhan Soydan, László Németh, László Szalay (2018)

Archivum Mathematicum

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Let $F_{n}$ denote the $n^{t h}$ term of the Fibonacci sequence. In this paper, we investigate the Diophantine equation $F_{1}^{p} + 2 F_{2}^{p} + \dots + k F_{k}^{p} = F_{n}^{q}$ in the positive integers $k$ and $n$ , where $p$ and $q$ are given positive integers. A complete solution is given if the exponents are included in the set ${1, 2}$ . Based on the specific cases we could solve, and a computer search with $p, q, k \leq 100$ we conjecture that beside the trivial solutions only $F_{8} = F_{1} + 2 F_{2} + 3 F_{3} + 4 F_{4}$ , $F_{4}^{2} = F_{1} + 2 F_{2} + 3 F_{3}$ , and $F_{4}^{3} = F_{1}^{3} + 2 F_{2}^{3} + 3 F_{3}^{3}$ satisfy the title equation.

On the Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

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We completely solve the Diophantine equations $x ² + 2^{a} q^{b} = y ⁿ$ (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).

Displaying similar documents to “On the diophantine equation x y - y x = c z ”

Displaying similar documents to “On the diophantine equation $x^{y} - y^{x} = c^{z}$ ”