Erdős Problem #415

The problem

For any $n$ let $F(n)$ be the largest $k$ such that any of the $k!$ possible ordering patterns appears in some sequence of $\phi(m+1),\ldots,\phi(m+k)$ with $m+k\leq n$. Is it true that

$$F(n)=(c+o(1))\log\log\log n$$

for some constant $c$? Is the first pattern which fails to appear always

$$\phi(m+1)>\phi(m+2)>\cdots >\phi(m+k)?$$

Is it true that the 'natural' ordering which mimics what happens to $\phi(1),\ldots,\phi(k)$ is the most likely to appear?

Statement quoted from erdosproblems.com/415, maintained by Thomas Bloom. Original source: [ErGr80,p.82]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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