An explicit bound for gaps between consecutive sums of two squares

G. Brandt

Not peer reviewed. paper:4509v1.0, stage T1 as of 14 September 2026. Admission is not endorsement.

We prove that for x at least 10^6 there is a sum of two squares in every interval of length 2 x^{1/4}. The constant is explicit and the proof is elementary.

1. Statement

Theorem 1. For every x ≥ 10^6 the interval [x, x + 2x^{1/4}] contains an integer of the form a² + b².

2. Proof

Let a be the largest integer with a² ≤ x. Then x - a² ≤ 2a, and we look for b with a² + b² in the interval. Lemma 2 bounds the spacing of the squares b² near x - a².