Date & Time: Friday, March 01, 2013, 16:00-17:00.
Title:Grimm's Conjecture and smooth numbers
Speaker:Prof. Shanta Laishram
Abstract: Let $g(n)$ be the largest positive integer $k$ such that there are distinct primes $p_i$ for $1\leq i\leq k$ so that $p_i |n+i$. This function is related to a celebrated conjecture of C.A. Grimm which is considered very difficult. In this talk, we establish upper and lower bounds for $g(n)$ by relating its study to the distribution of smooth numbers. Standard conjectures concerning smooth numbers in short intervals imply $g(n) =O(n^\epsilon)$for any $\epsilon >0$. We also prove unconditionally that $g(n) =O(n^\alpha)$ with $0.45<\alpha <0.46$. The study of $g(n)$ has some interesting implications for gaps between consecutive primes. This is a joint work with R. Murty.