Euler Totient Function Calculator

Calculator Input

Positive integer n

Example: 36

Optional comparison number

Checks gcd and coprime status.

Coprime display limit

Maximum allowed display is 500.

Batch values 5, 8, 10, 12, 36, 97

Separate values with commas, spaces, or new lines.

Calculate Totient

Example Data Table

Formula Used

Euler’s totient function counts positive integers not greater than n that are coprime to n.

φ(n) = n × Π(1 - 1 / p)

In this formula, p represents each distinct prime factor of n.

For n = 1, this calculator uses the standard convention φ(1) = 1.

Direct GCD Check

A number k is counted when gcd(k, n) = 1. This confirms that k and n share no common divisor except one.

Prime Power Rule

For a prime power p^a, the value is p^a - p^a-1. The product rule combines separate prime powers.

How to Use This Calculator

1. Enter a positive integer in the main input field.
2. Add a comparison number when you need a gcd test.
3. Set the coprime display limit for the residue preview.
4. Add batch values when you need many totient results.
5. Press the calculate button to show the answer above the form.
6. Use the CSV or PDF buttons to save the result.

Euler’s Totient Function in Number Theory

Euler’s totient function, written as phi n, counts positive integers from one to n that share no common factor with n except one. It is a core idea in elementary number theory. It supports modular arithmetic, fraction reduction, cyclic groups, and many cryptographic examples.

Why the Function Matters

The value is not just a count. It describes how many residues are invertible modulo n. When n is prime, every smaller positive integer is coprime to n. So phi of a prime p equals p minus one. Composite numbers need deeper checking because repeated prime factors change the count.

Prime Factor Method

The fastest practical method uses prime factorization. First split n into distinct prime factors. Then multiply n by one minus one over each different prime factor. Repeated powers are not repeated in this product. For example, twelve equals two squared times three. The formula gives twelve times one half times two thirds, which equals four.

Direct Coprime Method

A direct method checks each integer from one to n. It uses the greatest common divisor test. If gcd k and n equals one, k is counted. This method is easy to understand. It is slower for large n. It is useful for learning, verifying results, and showing the reduced residue system.

Using the Calculator

Enter a positive integer. Add optional batch values when you want many results. You may also enter a comparison number. The tool reports the gcd and whether the pair is coprime. Set a display limit for the coprime list. Large lists are shortened to keep the page readable.

Reading the Output

The result shows the prime factorization, formula substitution, totient value, and related counts. The coprime ratio compares phi n with n. A higher ratio means more residues are usable. The export buttons save the current result for notes, assignments, or teaching records.

Common Uses

Students use the function to solve modular problems. Teachers use it to explain primes and composites. Programmers use it when building number theory utilities. It is also connected to Euler’s theorem, which states that a to the power phi n is congruent to one modulo n when a and n are coprime. This helps proofs.

FAQs