A Recursive Construction of Coprime Triples with Small Radicals

We introduce a recursive construction of monotone sequences generated from products of powers of two distinct prime numbers. Starting from two rational numbers on opposite sides of 1, the construction recursively produces two inequality chains approaching 1 from below and above. We prove that both sequences converge to 1. Using these sequences, we construct infinitely many coprime triples (𝑎𝑖, 𝑏𝑖, 𝑐𝑖) satisfying 𝑎𝑖 + 𝑏𝑖 = 𝑐𝑖. We establish the estimate 𝑟𝑎d(𝑎𝑖𝑏𝑖𝑐𝑖)/𝑐𝑖 → 0, as 𝑖 → ∞. Thus, the construction provides infinite families of coprime triples whose radical is asymptotically negligible compared with 𝑐𝑖. The method is entirely elementary and is based on a recursive inequality structure rather than on existing approaches to the construction of abc-triples.