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The Phi Function Introduction Also known as the totient function, the phi function ?(n) is the number of positive integers smaller than n that are co-prime with n, where n is any positive integer. The term co-prime refers to numbers which do not share any common factors. In this case, it means all the positive integers smaller than n which do not have any of the same factors as n. The number 1 is said to be co-prime with all positive integers because it is a prime number and is only divisible by itself. One thing that needs to be understood before I start, however, is that to find out ?(n), we need to know the factors of n, so that we can check if any of the positive integers smaller than it are divisible by those factors. However, even these factors of n must be divisible by its prime factors and larger numbers...

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