Is the Euclidean algorithm optimal among its peers?

Much more is known about cε(a, b), but this simple-to-prove upper bound suggests the proper formulation of the Euclidean’s (worst case) optimality among its peers—algorithms from rem: Conjecture. If an algorithm α computes gcd(x, y) from rem with time complexity cα(x, y), then there is a rational number r > 0 such that for infinitely many pairs a > b > 1, cα(a, b) > r log2 a. Our main aim here is to prove the following relevant result: Theorem A. If a recursive programα decides the coprimeness relationx ? y from=,<,+, , iq and rem, then for infinitely many coprime pairs a > b > 1,