Filaseta 1992 gaps between squarefree numbers ii

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theorem: Filaseta and Trifonov's theorem that there is a constant c > 0 such that, for all sufficiently large x, the interval (x, x + c x^{1/5} log x] contains a squarefree number, proved by elementary means.

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Filaseta, M. and Trifonov, O., On gaps between squarefree numbers II. J. London Math. Soc. (2) 45 (1992), no. 2, 215-221 (DOI 10.1112/jlms/s2-45.2.215). The copy read for this card is the author's AMS-TeX typescript, not the journal's edition, and prints no copyright or license line on its first or last page; the author's publication list that lists the paper (https://people.math.sc.edu/filaseta/paperindex.html, read 2026-10-02) states no copyright, license or terms; the term is unstated.

The paper's single Theorem states that there is a constant c > 0 such that for all sufficiently large x the interval (x, x + c x^{1/5} log x] contains a squarefree number, improving the authors' earlier exponents 8/37 (elementary) and 3/14 (exponential sums) and the earlier work of Fogels, Roth, Richert, Rankin, Schmidt, Graham-Kolesnik and the two authors separately. The proof is elementary: writing S for the count of non-squarefree integers in (x, x+h] with h = c x^{1/5} log x, the contribution S_1 from small primes p <= h sqrt(log x) is less than (pi^2/6 - 1)h + pi(h sqrt(log x)), hence at most (2/3)h by the prime number theorem or a Chebyshev estimate, so it suffices to show the large-prime contribution S_2 is << c^sigma x^{1/5} log x for some sigma < 1, with an implied constant independent of c. That reduces to counting d in dyadic ranges for which d^2 has a multiple in (x, x+h], which the authors handle by first differences, divided (second) differences and Roth's modified first difference of x/u^2, the approximate value of the multiplier m with m u^2 in (x, x+h], rather than by exponential sums. Page numbers cited on the result page are the typescript's (1--9): the Theorem is on p. 1 and the proof runs over pp. 1--8.

Source: https://people.math.sc.edu/filaseta/paperindex.html.

Bears on. #208: the Theorem (p. 1) gives $s_{n+1}-s_n\le c\,s_n^{1/5}\log s_n$ for the squarefree numbers $s_1<s_2<\cdots$ and all large $n$, so the problem's first question holds for every $\epsilon>1/5$; it says nothing about $\epsilon\le1/5$ or about the second question.

Results.

• Theorem (p. 1): there is a constant $c>0$ such that for $x$ sufficiently large the interval $(x,x+cx^{1/5}\log x]$ contains a squarefree number.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.