Euler's Totient Function Calculator
This calculator allows you to compute the Euler's Totient function, which is a crucial concept in number theory. It is particularly useful for mathematicians, students, and educators needing to find the count of integers up to a given integer that are coprime with it.
Calculate Totient Function
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Data Source and Methodology
All calculations are based on the mathematical principles of Euler's Totient function. For more information, visit dCode Euler's Totient. All calculations strictly adhere to these mathematical formulas and data.
The Formula Explained
The formula for Euler's Totient function, φ(n), is given as: φ(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk) where p1, p2, ..., pk are the distinct prime factors of n.
Glossary of Terms
- n: The number for which you want to calculate the totient.
- φ(n): The count of integers up to n that are coprime with n.
- p1, p2, ..., pk: Distinct prime factors of n.
How It Works: A Step-by-Step Example
Let's calculate φ(9):
- The prime factors of 9 are 3.
- Therefore, φ(9) = 9 * (1 - 1/3) = 6.
Frequently Asked Questions (FAQ)
What is Euler's Totient function used for?
It is used in number theory to count the number of integers up to a given integer that are coprime with it.
Can Euler's Totient function be applied to any integer?
Yes, it can be applied to any positive integer.
What does coprime mean?
Two numbers are coprime if their greatest common divisor (GCD) is 1.
Is there a fast way to compute large totient values?
For large integers, factorization can be computationally intensive, but efficient algorithms exist for practical purposes.
Why is Euler's Totient function important?
It plays a key role in various areas of number theory and has applications in cryptography.