Erdős Problem #322

Definitions and mathematical statement

Shared notation.

Unless a section says otherwise, ${\mathbb{N}}=\{1,2,\ldots\}$, ${\mathbb{Z}}$ is the ring of integers, ${\mathbb{Q}},{\mathbb{R}},{\mathbb{C}}$ are the rational, real, and complex fields, $[N]=\{1,\ldots,\lfloor N\rfloor\}$, and $\log$ is the natural logarithm. For real functions of a parameter tending to infinity, $f\ll g$ and $f=O(g)$ mean $|f|\leq Cg$ eventually for some fixed $C$; $f\gg g$ reverses this comparison; $f=o(g)$ means $f/g\to0$; $f\sim g$ means $f/g\to1$; and $f\asymp g$ means both order bounds. A subscript on $O$ or $\ll$ specifies allowed constant dependence. The expression $n^{o(1)}$ in an upper bound means growth smaller than $n^\epsilon$ for each fixed $\epsilon>0$. Local definitions govern any exception, finite-field convention, or use of the same symbol for another object. An asymptotic claim is not automatically an effective algorithm.

The notation $1_A^{(k)}(n)$ means $k$-fold additive convolution: it counts ordered tuples $(x_1,\ldots,x_k)\in{\mathbb{N}}^k$ satisfying $x_1^k+\cdots+x_k^k=n$. Equal coordinates are allowed. The degree and the number of summands are both $k$. [S1] [S2]

Statement recorded in the supplied source.

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? $ [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: For every k>=4, prove or refute a fixed exponent c_k>0 occurring infinitely often, and sharpen the maximal-order behavior of the representation function. [S1]

Transcription note. Only unambiguous escape/formatting damage and nonmathematical serialization debris were repaired in the displayed source statement.

Short English statement

How large can the number of representations as a sum of k positive kth powers become, and does it exceed a fixed power of the target infinitely often?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 2048 (EP-322), statement and background. The embedded literature-review date is 2026-08-17; that is the archive’s review date, not a fresh certification. Dataset landing page: https://huggingface.co/datasets/ulamai/UnsolvedMath.

• S2 T. F. Bloom, Erdős Problems, Problem 322, https://www.erdosproblems.com/322. Reference from the supplied record; the live problem page was not independently checked for this compilation.

• S3 Kurt Mahler, Note on Hypothesis K of Hardy and Littlewood, Journal of the London Mathematical Society 11 (1936), 136-138,

  \nolinkurl{doi:10.1112/jlms/s1-11.2.136.}

  https://doi.org/10.1112/jlms/s1-11.2.136. Bibliographic information from the supplied record.

• S4 Paul Erdős, On the Representation of an Integer as the Sum of k k-th Powers, Journal of the London Mathematical Society 11 (1936), 133-136,

  \nolinkurl{doi:10.1112/jlms/s1-11.2.133.}

  https://doi.org/10.1112/jlms/s1-11.2.133. Bibliographic information from the supplied record.

• S5 Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244. Bibliographic information from the supplied record.

• S6 Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437. Bibliographic information from the supplied record.