কীভাবে Eulers Totient Function গণনা করবেন
Eulers Totient Function কি?
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.
সূত্র
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
ধাপে ধাপে নির্দেশিকা
- 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
সমাধান করা উদাহরণ
ইনপুট
φ(12)
ফলাফল
4 (coprime: 1,5,7,11)
ইনপুট
φ(7)
ফলাফল
6 (prime: all 1–6 are coprime)
সচরাচর জিজ্ঞাসা
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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