Eulers Totient Function ની ગણતરી કેવી રીતે કરવી
Eulers Totient Function શું છે?
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.
સૂત્ર
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
પગલું દ્વારા પગલું માર્ગદર્શિકા
- 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
ઇનપુટ
φ(12)
પરિણામ
4 (coprime: 1,5,7,11)
ઇનપુટ
φ(7)
પરિણામ
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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