Polycarpou’s theorem

Polycarpou’s theorem states that for any positive integers \(a\) and \(b\) that are coprime (meaning they have no common divisors other than 1), the inequality \(\frac{a + b}{ab} \leq \frac{2}{\text{max}(a, b)}\) always holds. This relation emphasizes how the sum of two coprime numbers, when scaled by their product, compares to twice the larger of the two. The theorem essentially provides a bound that illustrates the relationship between sum and product for coprime pairs, highlighting the interaction between their relative sizes and common factors.