On moments of gaps between consecutive square-free numbers

Let 6,: a, < a2 < ... be an infinite sequence of positive integers, and let dn = an ll an denote the n th gap. For our present purpose we regard any sequence d as interesting if its elements are so irregularly distributed that the gaps do not conform to any apparent pattern. Thus an arithmetic progression (a + bn: n = 0, 1,2,.. . } is uninteresting in our sense because all its gaps are equal to the common difference b, and so too is the sequence of squares {n2: n = 0, 1, 2,... } uninteresting because here the gaps form the arithmetic progression (1 + 2 n: n =0, 1,2,...}. Indeed, no sequence is interesting whose n th term an is given precisely by some algebraic formula, for the same is then true of the gaps dn. Given an interesting sequence (, there are various questions that one may ask about the fluctuation in size of its gaps, and the purpose of this article is to consider some of these questions in the context of three well-known sequences: (i) the primes, (ii) integers that are sums of two squares, and (iii) the squarefree numbers (integers that are products of distinct primes). I shall restrict myself to results that can be proved by elementary and simple arguments, and I shall only mention some of the deeper known results. You will get some impression of the difficulty of the subject if I add that even the deepest results now available fall far short of the likely ultimate truth.

For a positive square-free integer <TEX>$d$</TEX>, let <TEX>$t_d$</TEX> and <TEX>$u_d$</TEX> be positive integers such that <TEX>${\epsilon}_d=\frac{t_d+u_d{\sqrt{d}}}{\sigma}$</TEX> is the fundamental unit of the real quadratic field <TEX>$\mathbb{Q}(\sqrt{d})$</TEX>, where <TEX>${\sigma}=2$</TEX> if <TEX>$d{\equiv}1$</TEX> (mod 4) and <TEX>${\sigma}=1$</TEX> otherwise For a given positive integer <TEX>$l$</TEX> and a palindromic sequence of positive integers <TEX>$a_1$</TEX>, <TEX>${\ldots}$</TEX>, <TEX>$a_{l-1}$</TEX>, we define the set <TEX>$S(l;a_1,{\ldots},a_{l-1})$</TEX> := {<TEX>$d{\in}\mathbb{Z}|d$</TEX> > 0, <TEX>$\sqrt{d}=[a_0,\overline{a_1,{\ldots},2a_0}]$</TEX>}. We prove that <TEX>$u_d$</TEX> < <TEX>$d$</TEX> for all square-free integer <TEX>$d{\in}S(l;a_1,{\ldots},a_{l-1})$</TEX> with one possible exception and apply it to Ankeny-Artin-Chowla conjecture and Mordell conjecture.

We prove a modification as well as an improvement of a result of K. Ford, D. R. Heath-Brown and S. Konyagin concerning prime avoidance of square-free numbers and perfect powers of prime numbers.

Let [Formula: see text] be a positive integer and let [Formula: see text] be an odd prime. For [Formula: see text], we study the distribution of consecutive square-free numbers of the forms [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text]. In addition, we study the distribution of consecutive square-free primitive roots modulo [Formula: see text] of the forms [Formula: see text],[Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], respectively.

Klaus Friedrich Roth, 1925-2015

The square sieve and consecutive square-free numbers

We show that there exist infinitely many consecutive square-free numbers of the form n2 + 1, n2 + 2. We also establish an asymptotic formula for the number of such square-free pairs when n does not exceed given sufficiently large positive number.

A positive integer is square-full if each prime factor occurs to the second power or higher. Each square-full number can be written uniquely as a square times the cube of a square-free number. The perfect squares make up more than three-quarters of the sequence {si} of square-full numbers, so that a pair of consecutive square-full numbers is a pair of consecutive squares at least half the time, with

We prove that, for almost all D, the fundamental solution \({x_{0}+y_{0}\sqrt D}\) of the associated Pell equation x2−Dy2 = 1 is greater than D1.749···. We also show a strong link between this question and the error term in the asymptotic formula of the number of pairs of consecutive square-free numbers.

A Fourier-like transform is defined over a ring of quadratic integers modulo a prime number <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> in the quadratic field <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(\sqrt{m})</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> is a square-free integer. If <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</tex> is a Fermat prime, one can utilize the fast Fourier transform (FFT) algorithm over the resulting finite fields to yield fast convolutions of quadratic integer sequences in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R(\sqrt{m})</tex> . The theory is also extended to a direct sum of such finite fields. From these results, it is shown that Fourier-like transforms can also be defined over the quadratic integers in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R( \sqrt{m})</tex> modulo a nonprime Fermat number.