Number Theory Seminar

For the superelliptic curves of the form \[(x+1) \cdots(x+i-1)(x+i+1)\cdots (x+k)=y^\ell \] with $y \neq 0, k \geq 3$, $\ell \geq 2,$ a prime and for $i \in [2,k] \setminus \Omega$, we show that $\ell< e^{3^k}$. Here $\Omega$ denotes the interval $[p_\theta , (k-p_\theta))$, where $p_{\theta}$ is the least prime $\geq k/2$. Bennett and Siksek obtained similar bound for $i=1$ in a recent paper. This talk is based on a recent joint work with Laishram and Saradha.