Problem Statement
Let $s_1<s_2<\cdots$ be the sequence of squarefree numbers. Is it true that, for any $\epsilon>0$ and large $n$,\[s_{n+1}-s_n \ll_\epsilon s_n^{\epsilon}?\]Is it true that\[s_{n+1}-s_n \leq (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}?\]
Categories:
Number Theory
Progress
Erdős [Er51] showed that there are infinitely many $n$ such that\[s_{n+1}-s_n > (1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n},\]so this bound would be the best possible.In [Er79] Erdős says perhaps $s_{n+1}-s_n \ll \log s_n$, but he is 'very doubtful'.
Filaseta and Trifonov [FiTr92] proved an upper bound of $s_n^{1/5+o(1)}$. Pandey [Pa24] has improved this exponent to $1/5-c$ for some constant $c>0$.
Granville [Gr98] showed that $s_{n+1}-s_n\ll_\epsilon s_n^\epsilon$ for all $\epsilon>0$ follows from the ABC conjecture.
See also [489] and [145]. A more general form of this problem is given in [1101].
Source: erdosproblems.com/208 | Last verified: January 14, 2026