Alternatively have a look at the program.

## On some extensions of the Ailon-Rudnick Theorem

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,

## Counting multiplicatively dependent vectors of algebraic numbers

## On coincidences among quadratic fields generated by the Shanks sequence

## Strategies to solve congruence problems

We will review some of the classical strategies to solve congruence problems and discuss the limits to them. We will focus in estimating the number of solutions to \[ f(x,y) \equiv 0 \pmod p \quad 1\le x,y \le M \] where $f$ is some interesting function (polynomial, exponential, etc.). When $M$ is large, the classical approach on character sums/Fourier Analysis allow us to obtain asymptotics for this quantity. Nevertheless, there seems to be a barrier to this method at $M=p^{1/2}$ and new ideas, based on Additive Combinatorics, are required for the case when $M$ is small.

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