## 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.