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# Euler’s Totient Function

Updated on January 5, 2024
Article byWallstreetmojo Team
Edited byAshish Kumar Srivastav
Reviewed byDheeraj Vaidya, CFA, FRM

## What is Euler’s Totient Function?

Euler’s totient function is the mathematical multiplicative function that counts the positive integers up to the given integer, generally called ‘n,’ which is a prime number to ‘n.’ One may use the function to know the number of prime numbers that exist up to the given integer ‘n.’

### Key Takeaways

• Euler’s totient function is a mathematical multiplicative function that sums up the positive integers to a given integer’ n’. It is denoted as ‘ϕ(n)’ and is commonly used to determine the number of prime numbers up to ‘n’.
• This function provides insights into the properties and distribution of prime numbers, making it useful for analyzing number theory and cryptography.
• Euler introduced the totient function in 1763, initially denoting it with the Greek letter π, but later changed it to ϕ due to notation concerns.

### Explanation

Euler’s totient function one may use to know how many prime numbers are coming up to the given integer ‘n.’ It is also called an arithmetic function. Two things are important for an application or use of Euler’s totient function. One is that the gcd formed from the given integer ‘n’ should be multiplicative. The other is that the numbers of gcd should be the prime numbers only. The integer ‘n’ in this case should be more than 1. Calculating the Euler’s totient function from a negative integer is impossible. The principle, in this case, is that for ϕ(n), the multiplicators called m and n should be greater than 1. Hence, denoted by 1<m<n and gcd (m, n) = 1. Sign ϕ is the sign used to denote the totient function.

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### History

Euler introduced this function in 1763. Initially, Euler used the Greek π for denotation of the function, but because of some issues, his denotation of Greek π didn’t get recognition. And, he failed to give it the proper notation sign, i.e., ϕ. Hence, it cannot introduce the function. Further, ϕ was from Gauss’s 1801 Disquisitiones Arithmeticae. The function is also known as the phi function. But J. J. Sylvester, in 1879, included the term totient for this function because of its properties and uses. The different rules deal with different kinds of integers, such as if integer p is a prime number, then which rule to apply, etc. Euler frames all the rules as practicable. Therefore, one can use it even today while dealing with the same.

### Properties of Euler’s Totient Function

There are some different properties. Some of the properties of Euler’s totient function are as under:

• Φ is the symbol used to denote the function.
• The function deals with the prime numbers theory.
• The function is applicable only in the case of positive integers.
• For ϕ (n), one can find two multiplicative prime numbers to calculate the function.
• The function is a mathematical function and useful in many ways.
• If integer ‘n’ is a prime number, then gcd (m, n) = 1.
• The function works on the formula 1< m< n, where m and n are the prime and multiplicative numbers.
• In general, the equation is:

Φ (mn) = ϕ (m) * ϕ (n) (1- 1 / m) (1 – 1/ n)

For eg:
Source: Euler’s Totient Function (wallstreetmojo.com)

• T
• The function counts the number of positive integers less than the given integer, which is relatively prime numbers to the given integer.
• If the given integer p is prime, then ϕ (p) = p – 1
• If the power of p is prime, then if a = pis a prime power, then ϕ (pn) = pn – p ( n-1)
• ϕ (n) is not one – one
• ϕ (n) is not onto.
• ϕ (n), n > 3 is always even.
• ϕ( 10n) = 4 * 10 n-1

### Calculate Euler’s Totient Function

#### Example #1

Calculate ϕ (7)?

Solution:

ϕ ( 7 ) = (1,2,3,4,5,6) = 6

As all numbers are prime to 7, it made it easy to calculate the ϕ.

#### Example #2

Calculate ϕ ( 100 )?

Solution:

As 100 is a large number, it is time-consuming to calculate from 1 to 100 the prime numbers, which are prime numbers with 100. Hence, we apply the formula below:

• ϕ (100) = ϕ (m) * ϕ (n) (1- 1 / m) (1 – 1/ n)
• ϕ (100) = 2 2 * 2 5
• ϕ (100) = 2 2 * 2 5 * (1 – 1/2) * ( 1 – 1/5)
• = 100 * 1/2 * 4/5
• = 40

#### Example #3

Calculate ϕ (240)?

Multiples of 240 are 16*5*3 i.e. 2 4 * 5 * 3

• ϕ (240) = ϕ (m) * ϕ (n) (1- 1 / m) (1 – 1/ n)
• ϕ (240) = 2 4 * 5 * 3

if nM is not prime number the we use nm – nm-1

• = ( 2 4 – 2 (4-1) ) * (51 – 5 (1-1)) * (3 1 – 3 (1-1) )
• = (2 4 – 2 3) * (5 – 1) * (3 – 1)
• = 64

#### Example #4

Calculate ϕ ( 49 )?

• ϕ (49) = ϕ (m) * ϕ (n) (1- 1 / m) (1 – 1/ n)
• ϕ (49) = ϕ (7) * ϕ (7)
• = ( 71 – 7(1-1) ) * (71 – 7(1-1))
• = (7-1) * (7-1)
• = 6 * 6
• = 36

### Applications

The various applications are as under:

• One may use the function to define the RSA encryption system used for internet security encryption.
• One may use it in prime numbers theory.
• One may use it in large calculations also.
• One may use it in applications of elementary number theory.

### Conclusion

Euler’s totient function is useful in many ways. One may use it in the RSA encryption system for security purposes. The function deals with the prime number theory, and it is useful in the calculation of large calculations also. One may also use this function in algebraic calculations and elementary numbers. The symbol used to denote the function is ϕ, also called a phi function. The function consists of more theoretical use rather than practical use. The practical use of the function is limited. One can understand the function through various practical examples rather than theoretical explanations. There are various rules for calculating the Euler’s totient function, and different rules apply to different numbers. The function was first introduced in 1763. Due to some issues, it got recognition in 1784, and they modified the name in 1879. The function is universal and can be applied everywhere.

1. When to use Euler’s totient function?

Euler’s totient function is useful in various numbers theory and cryptography scenarios. It is commonly used in encryption algorithms, primality testing, and solving modular arithmetic problems. Euler’s totient function comes into play whenever there is a need to analyze the properties of prime numbers or determine the number of relatively prime integers to a given number.

2. What are the benefits and importance of Euler’s totient function?

The benefits of Euler’s totient function lie in its ability to provide insights into the distribution and properties of prime numbers. It allows mathematicians and cryptographers to understand the behavior of prime numbers, which is crucial in areas such as number theory, cryptography, and algorithm design. The function helps design secure encryption algorithms and plays a fundamental role in Euler’s theorem and Fermat’s little theorem.

3. What are the limitations of Euler’s totient function?

Although Euler’s totient function is a powerful mathematical tool, it does have certain limitations. One limitation is that computing the totient function for large numbers can be computationally expensive and time-consuming. The function assumes that the input number is a positive integer, so it may not directly apply to non-integer or negative values.

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