Date:
Wed, 06/04/2016 - 14:15 - 15:15
Let $a,b$ be multiplicatively independent positive integers and
$\varepsilon>0$. Bugeaud, Corvaja and Zannier (2003) proved that
$$
\gcd (a^n-1,b^n-1)\le \exp(\varepsilon n)
$$
for a sufficiently large $n$. Moreover, Ailon and Rudnick conjectured
that when $\gcd(a-1,b-1)=1$, then
$\gcd (a^n-1,b^n-1)=1$ infinitely often. Using finiteness of the number
of torsion points on curves, Ailon and Rudnick (2004) proved the
function field analogue of this conjecture, in a stronger form, that is,