Finding almost squares

Abstract

We study short intervals which contain an "almost square", an integer n that can be factored as n = ab with a, b close to √ n. This is related to the problem on distribution of n 2 α (mod 1) and the problem on gaps between sums of two squares.

Finding Almost Squares V

“…In a series of papers [1], [2], [3], [4], the author was interested in finding almost squares of either types in short intervals. In particular, given 0 Ä Â Ä 1 2 , we want to find "admissible" exponent i 0 (as small as possible) such that, for some constants C Â;i ; D Â;i > 0, the interval OEx D Â;i x i ; x C D Â;i x i contains a (Â; C Â;i )-almost square of type i (i D 1; 2) for all sufficiently large x.…”

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.…”

“…In [1], the author studied Question 1.2. Given 0 ≤ θ ≤ 1/2 and for all sufficiently large positive real number x, how short an interval around x, say [x−C x φ , x+C x φ ], is guaranteed to contain a (θ, C)-almost square.…”

“…Precise unconditional results are known for 0 ≤ θ ≤ 1/4 only and some conditional result is known for 1/4 < θ ≤ 3/10 (see [1] for details). It is conjectured that one can take φ = 1/2 − θ when 1/4 < θ < 1/2, and φ = , any small positive real number, when θ = 1/2.…”

Distance Between Arithmetic Progressions and Perfect Squares

“…What happens when N > d + 2 √ a? Retracing the steps in (1), this means that M > d. So there are more perfect squares in the interval [a, a + N d] than the common difference d of the arithmetic progression. However m 2 ≡ (m + d) 2 (mod d).…”