Kaip apskaičiuoti Eulers Totient Function

Kas yra Eulers Totient Function?

Euler's totient function φ(n) counts how many integers from 1 to n are coprime to n (share no common factor other than 1). It is fundamental in number theory and RSA encryption.

Formulė

φ(n) = n × ∏(1 − 1/p) for all prime factors p of n; for prime p: φ(p) = p−1

n: positive integer
φ(n): Euler totient of n — count of integers coprime to n

Žingsnis po žingsnio vadovas

1For prime p: φ(p) = p−1
2φ(pᵏ) = pᵏ−pᵏ⁻¹
3Multiplicative: φ(mn) = φ(m)φ(n) when gcd(m,n)=1
4φ(12) = φ(4)×φ(3) = 2×2 = 4

Worked Examples

Įvestis

φ(12)

Rezultatas

4 (coprime: 1,5,7,11)

Įvestis

φ(7)

Rezultatas

6 (prime: all 1–6 are coprime)

Frequently Asked Questions

Why is φ(n) important in cryptography?

φ(n) is essential to RSA encryption: the security depends on the difficulty of computing φ for large products of primes.

What does "coprime" mean?

Two numbers are coprime if their greatest common divisor (GCD) is 1. They share no common factor except 1.

What is φ(p) for a prime p?

φ(p) = p−1, because all numbers 1 to p−1 are coprime to p.

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