Expand all Collapse all | Results 1 - 2 of 2 |
1. CJM 2008 (vol 60 pp. 491)
A Multi-Frey Approach to Some Multi-Parameter Families of Diophantine Equations We solve several multi-parameter families of binomial Thue equations of arbitrary
degree; for example, we solve the equation
\[
5^u x^n-2^r 3^s y^n= \pm 1,
\]
in non-zero integers $x$, $y$ and positive integers $u$, $r$, $s$ and $n \geq 3$.
Our approach uses several Frey curves simultaneously, Galois representations
and level-lowering, new lower bounds for linear
forms in $3$ logarithms due to Mignotte and a famous theorem of Bennett on binomial
Thue equations.
Keywords:Diophantine equations, Frey curves, level-lowering, linear forms in logarithms, Thue equation Categories:11F80, 11D61, 11D59, 11J86, 11Y50 |
2. CJM 2001 (vol 53 pp. 897)
On Some Exponential Equations of S.~S.~Pillai In this paper, we establish a number of theorems on the classic
Diophantine equation of S.~S.~Pillai, $a^x-b^y=c$, where $a$, $b$ and
$c$ are given nonzero integers with $a,b \geq 2$. In particular, we
obtain the sharp result that there are at most two solutions in
positive integers $x$ and $y$ and deduce a variety of explicit
conditions under which there exists at most a single such solution.
These improve or generalize prior work of Le, Leveque, Pillai, Scott
and Terai. The main tools used include lower bounds for linear forms
in the logarithms of (two) algebraic numbers and various elementary
arguments.
Categories:11D61, 11D45, 11J86 |