Jinsi ya kukokotoa Eulers Totient Function

Eulers Totient Function ni nini?

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.

Fomula

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

Mwongozo wa Hatua kwa Hatua

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

Mifano Iliyotatuliwa

Ingizo

φ(12)

Matokeo

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

Ingizo

φ(7)

Matokeo

6 (prime: all 1–6 are coprime)

Maswali yanayoulizwa mara kwa mara

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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