What does Euler's totient function φ(n) count?

It counts the number of integers from 1 to n that are coprime to n — meaning they share no common factor with n other than 1.

What is the formula for φ(n)?

φ(n) = n × ∏ (1 − 1/p), where the product runs over each distinct prime factor p of n. For example, φ(12) = 12 × (1 − 1/2) × (1 − 1/3) = 12 × 1/2 × 2/3 = 4.

What is φ(p) for a prime number p?

For any prime p, φ(p) = p − 1, since every integer from 1 to p−1 shares no factor with a prime p.

φ(1) = 1 by convention, even though 1 has no prime factors — it's treated as coprime to itself as a special case.

Is Euler's totient function multiplicative?

Yes — for any two coprime integers m and n, φ(mn) = φ(m) × φ(n). This property, combined with the prime-power formula, is what makes the product formula work.

How large a value of n can this calculator handle?

This calculator supports n up to 1 trillion, using trial-division factorization up to √n, which stays fast even at that scale.

Why is Euler's totient function important in cryptography?

RSA key generation computes φ(n) = φ(p) × φ(q) = (p−1)(q−1) for two large secret primes p and q, then uses it to derive the private key — this is only tractable because the key generator knows p and q directly, not by factoring n itself.

How is this related to the Modular Multiplicative Inverse Calculator?

Euler's theorem states that a^φ(n) ≡ 1 (mod n) for any a coprime to n, which is the basis for one way to compute modular inverses: a's inverse is a^(φ(n)−1) mod n.

Euler's Totient Function Calculator — Count Integers Coprime to n

Our Euler's Totient Function Calculator computes φ(n), the number of integers from 1 to n that share no common factor with n, by finding n's prime factorization and applying the product formula φ(n) = n × ∏(1 − 1/p) over each distinct prime factor p — the same method used throughout number theory and RSA cryptography.

Quick Answer

φ(n) counts the integers from 1 to n coprime to n. Enter n below to see φ(n), computed from n's prime factorization via φ(n) = n × ∏(1 − 1/p).

Enter n, then click Calculate.

How to Use the Euler's Totient Function Calculator — φ(n) Online

1
Enter a positive whole number n.
2
Click 'Calculate' to see φ(n) along with n's full prime factorization.
3
Review the product formula breakdown showing exactly how φ(n) was derived.

Why Use Euler's Totient Function Calculator — φ(n) Online?

Euler's totient function, introduced by Leonhard Euler, counts how many integers from 1 to n are 'coprime' to n (share no common factor). It has a clean product formula once you know n's prime factorization: φ(n) = n × ∏(1 − 1/p), multiplying over each distinct prime factor p. This function is the mathematical heart of RSA cryptography — the security of RSA key generation depends on φ(n) being easy to compute when you know n's prime factors, but essentially impossible to compute (or use to factor n) when you don't.

Frequently Asked Questions

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