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Totient Function - mklabgr - 07-13-2026
Totient Function
Summary
Euler’s totient function, written as φ(n) is an important function in number theory that counts how many positive integers up to n are relatively prime to n meaning they share no common factors with it except 1. For example, $φ(24)=8$ because only eight numbers from 1 to 24 are coprime with 24.
The function provides a way to measure the “multiplicative freedom” of a number and is closely connected to prime factorization through the formula $φ(n)=n∏(1−1/p)$, where the product is taken over the prime factors of n. It has many interesting properties, such as always being even for n≥3 and satisfying the identity that the sum of φ(d) over all divisors d of n equals n.
Beyond pure mathematics, the totient function plays a central role in modern cryptography, especially in systems like RSA encryption. The article also explores its relationships with other important number-theoretic functions, including the Möbius function, and presents several mathematical identities and bounds that reveal the deeper structure of φ(n).
ARTICLE