Eulers Totient Function ಅನ್ನು ಹೇಗೆ ಲೆಕ್ಕ ಹಾಕುವುದು
Eulers Totient Function ಎಂದರೇನು?
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.
ಸೂತ್ರ
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
ಹಂತ-ಹಂತದ ಮಾರ್ಗದರ್ಶಿ
- 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
ಇನ್ಪುಟ್
φ(12)
ಫಲಿತಾಂಶ
4 (coprime: 1,5,7,11)
ಇನ್ಪುಟ್
φ(7)
ಫಲಿತಾಂಶ
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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