Euler's partition theorem

Euler’s partition theorem states that the number of ways to express a number as a sum of distinct positive integers (each used at most once) is equal to the number of ways to express the same number as a sum of positive integers where no part repeats more than times of a fixed shape, specifically, in the case of partitions into odd parts. Essentially, it reveals a surprising equivalence between two different ways of breaking down numbers into parts—one using only unique parts, and the other using parts with certain restrictions—highlighting a deep symmetry in the structure of numbers.