# The Phi function.

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Introduction

Maths Coursework

The Phi function

The phi function says that for any positive integer such as n the phi is Φ(n) and is defined as the number of positive integers, less than n which have no factor (other than 1) in common which means that they are co prime with n. If we take for example the phi of various numbers we will find that there is some relationship between them and I will investigate this relationship throughout.

I will also be investigating the results and the formulae for Φ(n2) and thus the formula for Φ(nx).

For example Φ(20) = 8

This was obtained by the following ways

First, list all the factors of 20 and all the numbers till 20.

1,2,4,5,10,20 These numbers can immediately be cancelled from the list of numbers till 20. The remaining numbers are 3,6,7,8,9,11,12,13,14,15,16,17,18,19. But the numbers 6,8,12,14,15,16,18 can also be cancelled out because one of their factors is the same as the factor for 20 and so cannot be considered. So from the following list

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 and so the final numbers remaining are 1,3,7,9,11,13,17 and 19 and soΦ(20) = 8

This way we can draw a table as is shown on the next page to find out the Φ values for all the numbers till 40. The table is shown on the next page. By using this table we will investigate further on the relationships of the values.

We will find the value of various numbers such as

Φ(3) = 2

Φ(8) = 4

Φ(11) = 10

Φ(24) = 8

Middle

6

36

12

6 x 2

7

49

42

7 x 6

8

64

32

8 x 4

9

81

54

9 x 6

10

100

40

10 x 4

11

121

110

11 x 10

From the above table we can see that the prime numbers have a definite pattern. They have a constant formula as we have seen earlier. Therefore prime numbers are the most suitable to be used for this investigation. We will now see the investigation for the values of Φ(p)2. p stands for prime.

The following table lists all the prime numbers and their squares and the phi values of those squares.

(p) | (p)2 | Φ(p)2 | |

2 | 4 | 2 | 2 x 1 |

3 | 9 | 6 | 3 x 2 |

5 | 25 | 20 | 5 x 4 |

7 | 49 | 42 | 7 x 6 |

11 | 121 | 110 | 11 x 10 |

We can deduce the formula. The formula will be Φ(p)2 = p(p-1). This we got because the Φ(p) = p-1 and so the p is multiplied to the formula and the so the above formula is obtained.

This formula has been explained logically and it can also be proved using sequences. The numbers that are taken are

(2,2) (3,6) (5,20) The number on the left is p and is regarded as x and the number on the right is the Φ(p2) for the corresponding value. This is substituted as y in the equation which is y = ax2 + bx +c. Thus there will be 3 equations once we substitute the 3 sets of numbers.

20 = 25a + 5b + c ……..1

6= 9a + 3b + c ……..2

2= 4a + 2b + c ……..3

We then subtract the second equation from the first one. The result will be

14 = 16a + 2b ……4

Then the second has to be subtracted from the third.

4 = 5a + b ………5

Now we can solve equation 4 and 5 simultaneously

14 = 16a + 2b

(4 = 5a + b) x –2

14 = 16a + 2b

-8 = -10a – 2b

6 = 6a

so a=1

Then we substitute a in the fourth equation

14 = 16 x 1 + 2b

-2 = 2b

b = -1

a=1 b = -1

Now the values for a and b have to substituted in the first equation

20=25 – 5 +c

20 = 20 +c

c=0

thus a=1, b = -1 and c=0 By entering these values into the equation y = ax2 + bx + c the final equation will be y = x2 –x . Thus the answer is

Φ(p)2 = p(p-1)

Now we will check this.

Φ(2)2 = 2(2-1)

Φ(4) = 2, 2(2-1) =2

So the above formula is correct.

So it can be said that

Φ(13)2 =13(13-1)

Φ(169) = 156

Therefore the formula has been proved.

Since we have to find out the formula for Φ(p)n, we also have to have to find out the values of Φ(p)3 . This we will do now.

(p) | (p)3 | Φ(p)3 | |

2 | 8 | 4 | 2 x 2 |

3 | 27 | 18 | 3 x 6 |

5 | 125 | 100 | 5 x 20 |

7 | 343 | 294 | 7 x 42 |

The formula for this would be Φ(p)3 = p x Φ(p)2. This is the same as p x p(p-1). This is because p has been multiplied to p2 to give p3 and thus will also be multiplied to p(p-1) to give p x p(p-1). This formula can also be derived by using the formula y = ax3 + bx2 + cx + d because this is for cubic functions but it will not be used because it will be longer and more difficult to understand. Thus it would be better to understand it logically.

Now we will check this formula with some numbers.

Φ(p)3 = p x p(p-1)

Φ(2)3 = 4(1)

Φ(8) = 4

This is true.

For another value

Φ(5)3 = 25(4)

Φ(125) = 100

This is also true.

This proves that Φ(p)3 = p2(p-1)

Now I will investigate the same way for Φ(p)4

(p) | (p)4 | Φ(p)4 | |

2 | 32 | 16 | 2 x 8 |

3 | 81 | 54 | 3 x 18 |

5 | 625 | 500 | 4 x 125 |

Conclusion

- Prime and even

Here we will check if Φ(mn) = Φ(m) x Φ(n) is true when m and n are prime and even.

Φ(2) x Φ(2) = Φ(4)

1 x 1 = 1

Φ(4) =1

This is not true.

Φ(3) x Φ(8) = Φ(24)

2 x 4 = 8

Φ(24) = 8

This is true.

Φ(5) x Φ(6) = Φ(30)

4 x 2 = 12

Φ(30) = 12

This is false.

Φ(4) x Φ(2) = Φ(8)

2 x 1 = 2

Φ(8) = 2

Thus this too is also false.

Overall Conclusion: The above formula is true only when 3 is used as one number and a number other than a multiple of three is used. For example

Φ(3) x Φ(6) = Φ(18)

2 x 2 = 4

Φ(18) = 4

This is false.

Φ(3) x Φ(10) = Φ(30)

2 x 4 = 8

Φ(30) = 8

This is true.

Φ(3) x Φ(8) = Φ(24)

2 x 4 = 8

Φ(24) = 8

This is also is true.

The above conclusion has been proved.

- Even and even

Here we will see when m and n in the formula are substituted by even numbers.

Φ(4) x Φ(4) = Φ(16)

2 x 2 = 8

Φ(16) = 8

This is false.

Φ(6) x Φ(4) = Φ(24)

2 x 2 = 4

Φ(24) = 8

This again is false.

Φ(8) x Φ(4) = Φ(32)

4 x 2 = 8

Φ(32) = 8

This is false.

Φ(2) x Φ(4) = Φ(8)

1 x 2 =2

Φ(8) = 2

This is false.

Overall conclusion: From the above pattern we can see that although there is no direct relation as such, we can understand that if 2 is multiplied to most of the numbers it will give the correct phi. So the formula for this is actually Φ(mn) = Φ(m) x Φ(n) x 2

For example

2 x Φ(4) x Φ(6) = Φ(24)

2 x 2 x 2 = 8

Φ(24) = 8

This is true.

2 x Φ(8) x Φ(2) = Φ(16)

2 x 4 x 1 = 8

Φ(16) =8

This is also true.

So we can see that the formula Φ(mn) = Φ(m) x Φ(n) x 2 is true only when m and n both are even.

This student written piece of work is one of many that can be found in our GCSE Phi Function section.

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