Results 101 to 110 of about 147 (145)

A Pure Exponential Diophantine Equation

Bulletin of Pure & Applied Sciences- Mathematics and Statistics, 2016
Let u be an even integer. In this paper we prove that if 4| u and u ≥208, then the equation (u3-3u)x + (3u2-1)y = (u2 +1)z has only the positive integer solution (x, y, z) = (2, 2, 3). This result basically determine all solutions of the equation.
Zhong Li, Wei-xun Li
openaire   +1 more source

Exponential Diophantine equations over function fields

Publicationes Mathematicae Debrecen, 1992
Let \(k\) be an algebraically closed field of characteristic zero and let \(k(t)\) be the field of rational functions over \(k\). Further, let \(\mathbb{K}\) be a finite extension of \(k(t)\). For given non-zero elements \(f_ 1,\ldots,f_ n,g\) of \(\mathbb{K}[X_ 1,\ldots,X_ n]\) \((n\geq 2)\), consider the equation \[ \sum^ n_{i=1}f_ i({\mathbf x ...
openaire   +2 more sources

Classification of Quantifier Prefixes Over Exponential Diophantine Equations

Mathematical Logic Quarterly, 1986
After Matijasevič had solved the \(10^{th}\) problem of Hilbert in 1970, it was natural to consider other similar problems. The \(10^{th}\) problem of Hilbert can be viewed as the problem of classification of the quantifier prefix \(\exists\exists\ldots\exists\) over diophantine equations (polynomial equations).
Jones, J. P., Levitz, H., Wilkie, A. J.
openaire   +2 more sources

Exponential Diophantine Equations

2019
This paper is a very gentle introduction to solving exponential Diophantine equations using the technology of linear forms in logarithms of algebraic numbers.
openaire   +1 more source

A note on ternary purely exponential diophantine equations

Acta Arithmetica, 2015
Summary: Let \(a,b,c\) be fixed coprime positive integers with \(\min\{a,b,c\}>1\), and let \(m=\max \{a,b,c\}\). 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 \(\max \{x,y,z\}1\).
Hu, Yongzhong, Le, Maohua
openaire   +1 more source

ON EXPONENTIAL DIOPHANTINE EQUATIONS CONTAINING THE EULER QUOTIENT

Bulletin of the Australian Mathematical Society, 2014
AbstractLet $a$ and $m$ be relatively prime positive integers with $a>1$ and $m>2$. Let ${\it\phi}(m)$ be Euler’s totient function. The quotient $E_{m}(a)=(a^{{\it\phi}(m)}-1)/m$ is called the Euler quotient of $m$ with base $a$. By Euler’s theorem, $E_{m}(a)$ is an integer.
openaire   +1 more source

The Undecidability of Exponential Diophantine Equations

1966
Publisher Summary This chapter focuses on the undecidability of exponential Diophantine equations. It is not known whether exponential Diophantine sets are necessarily Diophantine. However, it is known that every exponential Diophantine equation could be transformed mechanically into an equivalent ordinary Diophantine equation in more unknowns ...
openaire   +1 more source

Some conjectures in the theory of exponential Diophantine equations

Publicationes Mathematicae Debrecen, 2000
The author formulates a conjecture which implies Pillai's conjecture and a theorem of \textit{A. Schinzel} and \textit{R. Tijdeman} [Acta Arith. 31, 199-264 (1976; Zbl 0339.10018)] that for a polynomial with integer coefficients and at least two distinct roots, there are only finitely many perfect powers in its values at integral points.
openaire   +2 more sources

Exponential diophantine equations

Let \(\mathbb G\) be a commutative algebraic group over \(\mathbb C\), not containing any algebraic subgroup isomorphic to the additive group \(\mathbb G_ a\). Let \(\Gamma\) be a subgroup of \(\mathbb G(\mathbb C)\) of finite rank, that is, there is a finitely generated subgroup \(\Gamma'\) of \(\Gamma\) such that all elements of \(\Gamma/\Gamma ...
openaire   +2 more sources

Conjunctions of exponential diophantine equations over $${\mathbb {Q}}$$

Archive for Mathematical Logic
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
openaire   +1 more source