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Factors of a perfect square

Tsz Ho Chan — 2014

Acta Arithmetica

We consider a conjecture of Erdős and Rosenfeld and a conjecture of Ruzsa when the number is a perfect square. In particular, we show that every perfect square n can have at most five divisors between n - n ( l o g n ) 1 / 7 and n + n ( l o g n ) 1 / 7 .

Visible Points on Modular Exponential Curves

Tsz Ho ChanIgor E. Shparlinski — 2010

Bulletin of the Polish Academy of Sciences. Mathematics

We obtain an asymptotic formula for the number of visible points (x,y), that is, with gcd(x,y) = 1, which lie in the box [1,U] × [1,V] and also belong to the exponential modular curves y a g x ( m o d p ) . Among other tools, some recent results of additive combinatorics due to J. Bourgain and M. Z. Garaev play a crucial role in our argument.

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