Euler's φ (phi) Function counts the number of positive integers not exceeding n and relatively prime to n. Traditionally, the proof involves proving the φ function is multiplicative and then proceeding to show how the formula arises from this fact. We ignore this fact, at least directly, and show a practical and sound method to calculate φ. We offer a proof of the closed form formula for this function relying on similar, but subtly different counting techniques.
Prof. Harold Reiter, Dept of Mathematical Sciences, University of North Carolina Charlotte
Chorge, Shashank and Vargas, Juan
"Proof of Euler's φ (Phi) Function Formula,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 14
, Article 6.
Available at: http://scholar.rose-hulman.edu/rhumj/vol14/iss2/6