Tsz Ho Chan

On sets of integers, none of which divides the product of k others

This paper is a continuation of previous work by Gyori, Sarkozy, and the author, concerning the maximal number of integers that can be selected from {1,2,...,N} so that none of them divides the product of k others.

SQUAREFULL NUMBERS IN ARITHMETIC PROGRESSIONS

In this paper, we study squarefull numbers in arithmetic progressions. We find the least such squarefull number by Dirichlets hyperbola method as well as Burgess bound on character sums. We also obtain a best possible almost all result via a large sieve inequality of Heath-Brown on real characters.

Finding Almost Squares V

Abstract An almost square of type 2 is an integer n that can be factored in two different ways as n = a 1 b 1 = a 2 b 2 with a 1, a 2, b 1, . In this paper, we continue the study of almost squares of type 2 in short intervals and improve the 1/2 upper bound. We also draw connections with almost squares of type 1.

Common factors among pairs of consecutive integers

In this paper, we study products of consecutive integers and prove an optimal bound on their greatest common factors. In contrast, we obtain a modest upper bound for the greatest common factor in the perfect square situation. Using this and an upper bound on the size of solutions to hyperelliptic curves, we prove a gap principle when a2(a2 + 1) divides b2(b2 + 1) with some additional restrictions. We also obtain a stronger gap principle under the abc conjecture.

Factors of almost squares and lattice points on circles

In this paper, we consider a conjecture of Erdos and Rosenfeld and a conjecture of Ruzsa when the number is an almost square. By the same method, we consider lattice points of a circle close to the x-axis with special radii.

TWIN SQUAREFUL NUMBERS

A number is squareful if the exponent of every prime in its prime factorization is at least two. In this paper, we give, for a fixed

A NOTE ON PRIMES IN SHORT INTERVALS

l

Finding almost squares VI

, the number of pairs of squareful numbers

SHORTEST DISTANCE IN MODULAR HYPERBOLA AND LEAST QUADRATIC NON-RESIDUE

n

TWIN CUBEFULL NUMBERS

,