Erdős–Straus conjecture for expressing 4/n as a sum of three unit fractions

Prove that for every integer n ≥ 2 there exist positive integers x, y, and z such that 4/n = 1/x + 1/y + 1/z.

Background

The Erdős–Straus conjecture (ESC) asserts that the rational number 4/n can always be decomposed into a sum of three unit fractions for any integer n ≥ 2. The paper reviews known partial results and emphasizes that the hardest remaining cases are primes congruent to 1 modulo 4 (Pythagorean primes), motivating the subsequent analysis and parametrizations.

The author splits the problem into two solution types (A and B), corresponding to whether one or two of the denominators are multiples of n, and develops equivalent Diophantine formulations intended to better understand solvability specifically for Pythagorean primes. The conjecture remains unproven in general.