Erdős Problem #675

declared status 'open'. Formalized: no. 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 $1\leq a\leq n$,$$a\in A\quad\textrm{ if and only if }\quad a+t_n\in A.$${UL} {LI}Does the set of the sums of two squares have the translation property?{/LI} {LI}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?{/LI} {LI}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 $c>0$?{/LI} {/UL} Prize: no. Tags: number theory.

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