Lethbridge Number Theory and Combinatorics Seminar: Ram Murty

A number $n$ is called squarefull if for every prime $p$ dividing $n$, we have $p^2$ also dividing $n$.

 

Erdos conjectured that the number of pairs of consecutive squarefull numbers $(n, n+1)$ with $n < N$ is at most $(log N)^A$ for some $A >0$.

 

This conjecture is still open.  We will show that the abc conjecture implies this number is at most $N^e$ for any $e>0$.  We will also discuss a related conjecture of Ankeny, Artin and Chowla on fundamental units of certain real quadratic fields and discuss its connection with the Erdos conjecture.

 

This is joint work with Kevser Aktas.