Chapin's Theorem

Chapin's Theorem addresses the problem of representing positive integers as sums of two specific types of numbers called *k*-th powers (numbers like 1, 8, 27 for cubes, etc.). It states that under certain conditions on the numbers involved, every large enough integer can be expressed as a sum of one *k*-th power and another *k*-th power that are coprime (share no common factors). This theorem helps us understand how complex numbers can be broken down into combinations of more fundamental power-sums, highlighting the structure and limits of such representations in number theory.