Is the Euclidean Algorithm Optimal Among its Peers?

The Euclidean algorithm on the natural numbers ℕ = {0,1,…} can be specified succinctly by the recursive program where rem( a, b ) is the remainder in the division of a by b , the unique natural number r such that for some natural number q , It is an algorithm from (relative to) the remainder function rem, meaning that in computing its time complexity function c ε (a, b) , we assume that the values rem( x, y ) are provided on demand by some “oracle” in one “time unit”. It is easy to prove that Much more is known about c ε (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: C onjecture . 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 log 2 a .