Euler's Totient/Phi Function (step 4)

Leonhard Euler investigated properties of numbers and prime number distribution, defining the phi (φ) function to measure a number’s “breakability” through its coprime relationships.

Computing Euler’s totient function proves computationally hard for arbitrary integers, creating computational asymmetry exploitable for cryptographic security.

The totient function’s core operation counts integers coprime to a target number—those sharing no common factors except one, called relatively prime numbers.

Euler’s totient function possesses the multiplicative property: φ(a × b) = φ(a) × φ(b), enabling efficient computation when factorization is known.

Prime numbers exhibit the simplest possible totient behavior, with φ(p) = p - 1 providing instant computation without factorization or coprimality testing.

Plotting Euler’s totient function φ(n) over integers from 1 to 1,000 reveals dramatically different behaviors for primes versus composites.