Problem 675

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 $c>0$ ?

Source: erdosproblems.com/675

Number theory

Wiki page

Status

Open

No claim settles this problem.

Open, the site's label (proof-claims thread accessed 2026-10-06). The site notes that elementary sieve theory gives the squarefree numbers the translation property, and that Brun's sieve gives it to the integers divisible by no member of a set $B$ of pairwise coprime integers with $\sum_{b<x}1/b=o(\log\log x)$ . A partial proof claim posted to the site's proof-claims tab on 27 July 2026 by Liam Price, using GPT 5.6 Sol Pro, answers the second question yes for a partition of the primes into parts of any prescribed proportions; it is recorded on its claim page (Price, 2026) and not adopted here. The first question and the growth questions are not addressed by it. The discussion thread carries a pending partial claim on the growth question: Boon Suan Ho's note of 18 April 2026, found with GPT-5.4 Pro, proves $t_n>\exp(n^c)$ for every $c<25/72$ and all large $n$ (its claim page (Ho, 2026)). A comment of 29 April 2026 raising the exponent to $3/8$ was questioned in the thread and has no write-up. Yu Leon Liu's note of 9 May 2026, found with OpenAI's Codex, shows that every shift for the sums of two squares exceeds $\exp(n^c)$ for $c<1/10$ . It does not decide whether that set has the translation property, so it settles no question and has no claim page.

Er79: Erdős, P., Some unconventional problems in number theory. Math. Mag. 52 (1979), no. 2, 67--70; item 5, on translation properties, printed p. 69, poses the three questions of the statement and states that the squarefree numbers, and the integers avoiding multiples of pairwise coprime $b_i$ with $\sum1/b_i<\infty$ , have the property, and that by Brun's method the weaker condition $\sum_{b_i<x}1/b_i=o(\log\log x)$ suffices. Library home: Erdős 1979.

Problem 675

Status

Proof claims

Discussion