# On Gaps between Squarefree Numbers

• Michael Filaseta
• Ognian Trifonov
Chapter
Part of the Progress in Mathematics book series (PM, volume 85)

## Abstract

A squarefree number is a positive integer not divisible by the square of an integer > 1. We investigate here the problem of finding small h = h(x) such that for x sufficiently large, there is a squarefree number in the interval (x,x + h]. This problem was originally investigated by Fogels [3]; he showed that for every ∈ > 0, h = x 2/5+∈ is admissible. Later Roth [9] reported elementary arguments of Davenport and Estermann showing respectively that one can take hx 1/3 and hx 1/3(log x)-2/3 for sufficiently large choices of the implied constants. Roth then gave an elementary proof that h = x 1/4+ ∈ is admissible, and by applying a result of van der Corput, he showed that one can take hx 3/13 (log x)4/13Nair [6] later noted that the elementary proof could be modified to omit the ∈ in the exponent to get that hx 1/4 is admissible, and more recently the first author [1] showed that one could obtain the result hx 3/13 by elementary means. Using further exponential sum techniques, Richert [8], Rankin [7], Schmidt [10], and Graham and Kolesnik [4] obtained the improvements hx 2/9log x, h = x θ +€ where θ = 0.221982…, θ = 109556/494419 = 0.221585…, and θ = 1057/4785 = 0.2208986…, respectively. The authors investigated the problem further.

## Keywords

Elementary Proof Divided Difference Implied Constant Prime Number Theorem Elementary Argument
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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