Problem 1004

Let $c>0$ . If $x$ is sufficiently large then does there exist $n\leq x$ such that the values of $\phi(n+k)$ are all distinct for $1\leq k\leq (\log x)^c$ , where $\phi$ is the Euler totient function?

Source: erdosproblems.com/1004

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open. The one claim recorded, the anonymous note posted on 2026-04-29, is partial: it asserts the block statement for every fixed $c<2$ in the almost-all form and says nothing about $c\ge2$ . The site labels the problem OPEN (page last edited 12 April 2026).

EPS87: Erdős, Paul and Pomerance, Carl and Sárközy, András, On locally repeated values of certain arithmetic functions. III. Proc. Amer. Math. Soc. (1987), 1-7.
Anon26: A partial result on distinct consecutive values of Euler's function, five-page note with no author line, linked from forum post 6051 by the account aditya, posted at 18:09 on 29 April 2026 (accessed 2026-09-07); the post attributes the result to Gpt 5.5 pro. Retained at the totient-blocks native-note lead as an unreviewed candidate and recorded as a partial claim.

Problem 1004

Status

Proof claims

Discussion