Gaps Between Sums of Two Squares

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.

A sum of two squares means $a^2+b^2$ with $a,b\in{\mathbb{Z}}$, including a zero summand. The asymptotic interval question concerns sufficiently large real $X$ and the interval with endpoints $X-X^{1/4}/10$ and $X$. The constant $1/10$ is part of the target, not an unspecified positive constant. [S1] [S2]

Statement recorded in the supplied source.

Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$? [S1]

Scope and interpretation.

The source background concerns a large-$X$ gap bound. This is the asymptotic reading; the wording always is not used to assert the false small-$X$ version. [S1]

Residual task reported by the archive.

The embedded review (2026-08-17) records: Improve the uniform one-sided constant to 1/10, or prove any o(X^{1/4}) worst-case gap bound. [S1]

Short English statement

Must every sufficiently large number have a sum of two squares within one tenth of its fourth root immediately below it?

Sources

• S1 ULAM.AI, UnsolvedMath, supplied problems.json, record 127 (GREEN-039), 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 Alexander Kalmynin, Intervals between numbers that are sums of two squares, Mathematika 65 (2019), 1018-1032. https://arxiv.org/abs/1706.07380. Bibliographic information from the supplied record.

• S3 Ben Green, 100 Open Problems, current Problem 66. https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf. The author-hosted document was inspected on 27 September 2026; the archive’s local code is not a problem-number locator in this paper.