The phi function.

Authors Avatar


The phi function.

Introduction.

Phi is a letter in the Greek alphabet.  In higher mathematics:

  • The upper case form of phi, (Φ) means an angle function
  • The lower case form, φ , means angle mathematics and golden ratio mathematics

For any positive integer n, the Phi Function f (n) is defined as the number of positive integers less than n, which has no factor (other than 1) in common (are co-prime) with n.

In this piece of work I will be investigating several instances involving phi and prime numbers. Patterns between numbers that are prime, co prime involving phi will be found.

PART 1.

  1. Find the values of:
  1. Φ (3)
  2. Φ (8)
  3. Φ (11)
  4. Φ (24)

Solution

  1. Φ (3) = 2, 1.

Φ (3) = 2

This is because there are only 2 positive integers less than 3 which have no common factors with 3 other than 1.

  1. Φ (8) = 7, 5, 3, 1.

Φ (8) = 4

This is because there are 4 positive integers less than 8 which have no common factors with 8 other than 1.

  1. Φ (11) = 10, 9, 8, 7, 6, 5, 4, 3, 2, 1.

Φ (11) = 10

This is because there are 10 positive integers less than 11 which have no common factors with 11 other than 1.

  1. Φ (24) = 23, 19, 17, 13, 11, 7, 5, 1.

Φ (24) = 8

This is because there are 8 positive integers less than 24 which have no common factors with 24 other than 1.

  1. Obtain the Phi Function for at least 5 positive integers of your own choice.

I intend to find the Phi Function of the following:

  1. Φ (5) because it is a prime number.
  2. Φ (10) because its an even number.
  3. Φ (15) because its an odd number
  4. Φ (16) because it’s squared
  5. Φ (27) because its cubed

Solution

  1. Φ (5) = 4, 3, 2, 1.

Φ (5) = 4

  1. Φ (10) =9, 7, 3, 1.

Φ (10) = 4

  1. Φ (15) = 14, 13, 12, 11, 8, 7, 4, 2, 1.

Φ (15) = 9

  1. Φ (16) = 15, 13, 11, 9, 7, 5, 3, 1.

Φ (16) = 8

  1. Φ (27) = 26, 25, 23, 22, 20, 19, 17, 16, 14, 13, 11, 10, 8, 7, 5, 4, 2, 1.

Φ (27) = 18

I noticed that the Phi of 3, 5, and 11, which are all prime numbers is themselves minus one. So when n is a prime number Φ (n) = n – 1

PART 2.

  1. Check that:
  1. Φ (7 x 4) = Φ (7) x Φ (4)

Solution:

Φ (7 x 4) = Φ (7) x Φ (4)

Join now!

Φ (7 x 4)

= Φ (28)

Φ (28) = 27, 25, 23, 19, 17, 15, 13, 11, 9, 5, 3, 1.

= Φ (28) = 12

Φ (7)

Φ (7) = 6, 5, 4, 3, 2, 1.

Φ (7) = 6

Φ (4)

Φ (4) = 3, 1.

Φ (4) = 2

Φ (7) x Φ (4)

= 6 x 2

= 12

As the answer gotten from Φ (7) x Φ (4) is equal to that of Φ (7 x 4), the equation Φ (7 x 4) = Φ (7) x Φ (4) is ...

This is a preview of the whole essay