Erdős Problem #675

The problem

We say that $A\subset \mathbb{N}$ has the translation property if, for every $n$, there exists some integer $t_n\geq 1$ such that, for all $1\leq a\leq n$,

$$a\in A\quad\textrm{ if and only if }\quad a+t_n\in A.$$

Does the set of the sums of two squares have the translation property? If we partition all primes into $P\sqcup Q$, such that each set contains $\gg x/\log x$ many primes $\leq x$ for all large $x$, then can the set of integers only divisible by primes from $P$ have the translation property? If $A$ is the set of squarefree numbers then how fast does the minimal such $t_n$ grow? Is it true that $t_n>\exp(n^c)$ for some constant $c>0$?

Statement quoted from erdosproblems.com/675, maintained by Thomas Bloom. Original source: [Er79]. 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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