Saradha Natarajan
TIFR
March 23, 2017
Erdos- Selfridge superelliptic curves and their rational points: Erdos 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, $\ell < k$ holds. This is a joint work with Pranabesh Das and Shanta Laishram.
TIFR
March 23, 2017
Erdos- Selfridge superelliptic curves and their rational points: Erdos 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, $\ell < k$ holds. This is a joint work with Pranabesh Das and Shanta Laishram.