Euler’s Totient Function

What is Euler’s Totient Function?

Euler’s Totient function is the mathematical multiplicative functions which count the positive integers up to the given integer generally called as ‘n’ that are a prime number to ‘n’ and the function is used to know the number of prime numbers that exist up to the given integer ‘n’.

Explanation

To know how many prime numbers are coming up to the given integer ‘n’ Euler’s Totient Function is used. It is also called an arithmetic function. For an application or use of Euler’s Totient function, two things are important. One is that the gcd formed from given integer ‘n’ should be multiplicative to each other, and the other is the numbers of gcd should be the prime numbers only. The integer ‘n’ in this case should be more than 1. From a negative integer, it is not possible to calculate the Euler’s Totient Function. 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 Totient Function.

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 the recognition. And he failed to give it the proper notation sign i.e., ϕ. Hence the function cannot be introduced. Further, ϕ was taken from the Gauss’s 1801 Disquisitiones Arithmeticae. The function is also termed as phi function. But J. J. Sylvester, in 1879, included the term totient for this function because of properties and the uses of the functions. The different rules are framed to deal with different kinds of integers given like if integer p is a prime number, then which rule to be applied, etc. all the rules are framed by Euler are practicable and can be used even today while dealing with the same.

Properties of Euler’s Totient Function

There are some of the different properties. Some of the properties of Euler’s totient function is 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), two multiplicative prime numbers are to be found 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 numbers and multiplicative numbers.
  • In general, the equation is

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

Euler’s-Totient-Function

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  • The function basically counts the number of positive integers less than the given integer, which is relatively prime numbers to the given integer.
  • If given integer p is prime then ϕ (p) = p – 1
  • If the power of p is prime then, if a = pn is 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, hence it made it easy to calculate the ϕ.

Example #2

Calculate ϕ ( 100 )?

Solution:

As 100 is large number hence 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:

  • The function is used for defining RSA encryption system used for internet security encryption.
  • Used in prime numbers theory.
  • Used in large calculations also.
  • Used in applications of elementary number theory.

Conclusion

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

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