The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper ...The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper bounds have been found for this problem. Here, we provide a sharp upper bound for a function which has a direct relation to the numbers whom the greatest common divisor we are trying to calculate. We mainly use some features of Fibonacci numbers as our tools.展开更多
文摘The problem of determining the number of steps needed to find the greatest common divisor of two positive integers by Euclidean algorithm has been investigated in elementary number theory for decades. Different upper bounds have been found for this problem. Here, we provide a sharp upper bound for a function which has a direct relation to the numbers whom the greatest common divisor we are trying to calculate. We mainly use some features of Fibonacci numbers as our tools.
