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 D a 1 b 1 D a 2 b 2 with a 1 , a 2 , b 1 , b 2 p n. 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.

Perfect squares have at most five divisors close to its square root

“…Suppose not, say µ i = µ j for some 1 ≤ i < j ≤ r. Then µ i x i y i = 2N = µ j x j y j implies x i y i = x j y j = 2N µi . Numbers like 2N µi that can be factored as x i y i and x j y j with x i close to y i and x j close to y j are called almost squares of type 2 and have been studied by the author in [1], [2] and [3] for example. If…”

Finding almost squares IV

“…In [2] and [3], the author studied almost squares of type 2 (an integer n with two distinct representations n = a 1 b 1 = a 2 b 2 where a 1 , b 1 , a 2 , b 2 are close to √ n), and was led to the question of finding an integer close to x which is divisible by some integer in another interval. More specifically, Question 1.5.…”