Erdős Problem 675

### What is the conjecture

https://www.erdosproblems.com/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 $c>0$?

Status: open

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

The issue names no file, test, or entry point and presents three related mathematical questions. Start by inspecting existing formalized number-theory conjectures in the repository and determine which question can be stated in Lean. Done means a reviewed formal statement is added with any required supporting definitions or lemmas.

Written by the indexing model from the issue text.