Problem 334

Erdős Problem #334: declared status 'open'. Formalized: no. Find the best function $f(n)$ such that every $n$ can be written as $n=a+b$ where both $a,b$ are $f(n)$-smooth (that is, are not divisible by any prime $p>f(n)$.) Current best: This is known, and the best bound is due to Balog [Ba89] who proved that$$f(n) \ll_\epsilon n^{\frac{4}{9\sqrt{e}}+\epsilon}$$for all $\epsilon>0$. Prize: no. OEIS: A045535, A062241. Tags: number theory.