Problem 322

Erdős Problem #322: declared status 'open'. Formalized: no. Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers. What is the order of growth of $1_A^{(k)}(n)$, i.e. the number of representations of $n$ as the sum of $k$ many $k$th powers? Does there exist some $c>0$ and infinitely many $n$ such that$$1_A^{(k)}(n) >n^c?$$ Current best: Erd\H{o}s believed Hypothesis $K$ fails for all $k\geq 4$, but this is unknown. Hardy and Littlewood made the weaker Hypothesis $K^*$ that for all $N$ and $\epsilon>0$$$\sum_{n\leq N}1_A^{(k)}(n)^2 \ll_\epsilon N^{1+\epsilon}.$$Erd\H{o}s and Graham remark: 'This is probably true but no doubt very deep. However, it would suffice for most applications.' Independently Erd\H{o}s [Er36] and Chowla proved that for all $k\geq 3$ and infinitely many $n$$$1_A^{(k)}(n) \gg n^{c/\log\log n}$$for some constant $c>0$ (depending on $k$). Prize: no. OEIS: A025418, A025456. Tags: number theory, powers.