Colloquium abstracts

Saradha Natarajan
TIFR
March 23, 2017

ErdH{o}s- Selfridge superelliptic curves and their rational points:  Erd{H o}s and Selfridge showed a remarkable result in 1975 that a product of two or more consecutive positive integers can never be a perfect power. In other words, the equation $$ (x+1) cdots (x+k)=y^ell$$ with $k geq 2$, $ell geq 2$ has no integral solution. We may look at it as a super elliptic curve and ask for rational solutions in $x$ and $y.$ In a recent paper, Bennet and Siksek showed that if a rational solution exists then $ell< e^{3^k}.$ In this talk, we consider some variations of the above curve and show that similar bound for $ell$ is valid. Further when $k$ is small, $ell