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  • Level: GCSE
  • Subject: Maths
  • Word count: 2134

The phi function.

Extracts from this document...

Introduction

Charlene Asare

Mathematics Coursework 2

Charlene Asare

Ashbourne College

Mathematics Coursework

April 2003

The phi function.

Ф


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.

...read more.

Middle

Φ (3 x 7) = x Φ (3) x Φ (7)

Φ (3 x 7)

= Φ (21)

Φ (21) = 20, 19, 17, 16, 13, 11, 10, 8, 5. 4, 2, 1.

= Φ (21) = 12

Φ (3)

= Φ (3) = 2,1.

Φ (3) = 2

Φ (7)

= Φ (7) = 6, 5, 4, 3, 2, 1.

Φ (7) = 6

As the answer gotten from Φ (21) is equal to that of Φ (3) x Φ (7), the equation Φ (n x m) = Φ (n) x Φ (m), where n = 3 and m = 7 is correct.

12 integers = 2 integers x 6 integers

12 integers = 12 integers.

Solution

For my third choice I will use the numbers 8 and 4 where n = 8 and 4= m.

Φ (n x m) = Φ (n) x Φ (m)

Φ (8 x 4) = Φ (8) x Φ (4)

Φ (8 x 4)

=  Φ (32)

Φ (32) = 31, 29, 27, 25, 23, 21, 19, 17, 15, 13, 11, 9, 7, 5, 3, 1.

Φ (32) = 16

Φ (8)

Φ (8) = 7, 5, 3, 1

= Φ (8) = 4

Φ (4)

Φ (4) = 3, 1

Φ (4) = 2

As the answer gotten from Φ (32) is not equal to that of Φ (4) x Φ (8), the equation Φ (n x m) = Φ (n) x Φ (m), where n = 8 and m = 4 is incorrect.

16 integers ≠ 4 integers x 2 integers

16 integers ≠ 8 integers

PART 3.

In some cases Φ (n x m) = Φ (n) x Φ (m) whilst in other cases this is not so. Investigate this situation.

Investigation:

From the results gotten in Part 2 its very obvious that in cases where the variables n and m have common factors the equation:

Φ (n × m) = Φ (n) × Φ (m);

does not work. In cases in which the variables n and m had no common factors the equation:

Φ (n × m) = Φ (n) × Φ (m);

worked. With this information I can draw a very accurate prediction.

Prediction:

...read more.

Conclusion

Analysis

A multiplicative function is a function f such that f(a × b) = f(a) × f(b). Is phi multiplicative?To an extent it is as in all the sets of numbers that had common factors the equation Φ (n × m) = Φ (n) × Φ (m) did not work.

Conclusion

With all the problems and examples I have solved in relation to this I can conclude that;

Φ (n x m) = Φ (n) x Φ (m)

where n is a prime number and m is any number at all.

PART 4.

If p and q are prime investigate: Φ (pmqn)

The equationΦ (pmqn) = Φ (pm) x Φ (qn) and so with the knowledge gotten from part 3 I can predict that Φ (pmqn) = Φ (pm) x Φ (qn) will be true where p and q are prime. to further illustrate this I will do a few examples.

Example 1.

P= 2, q = 3 ; m = 3, n = 2

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (23 x 32) = Φ (23)xΦ (32)

Φ (8 x 9) = Φ (8)xΦ (9)

Φ (72) = Φ (8) xΦ (9)

24 = 4 x 6

24 = 24

Example 2.

P= 7, q = 5 ; m = 4, n = 6

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (54 x 71) = Φ (54)xΦ (71)

Φ (625 x 7 ) = Φ (625)xΦ (7)

Φ (4375) =  Φ (625) xΦ (7)

3000 = 500 x 6

3000 = 3000

Example 3

P= 5, q = 11 ; m = 1, n = 3

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (51 x 113) = Φ (51)xΦ (113)

Φ (5 x 1331) = Φ (5)xΦ (1331)

Φ (6655) =  Φ (5) xΦ (1331)

4840 = 4 x 1210

4840 = 4840

Example 4

P= 13, q = 2 ; m = 2, n = 4

Φ (pm x qn)

Φ (pmqn) = Φ (pm)xΦ (qn)

Φ (132 x 24) = Φ (132)xΦ (24)

Φ (169 x 16 ) = Φ (169)xΦ (16)

Φ (2704) =  Φ (169) xΦ (16)

1248 = 156 x 8

1248 = 1248

...read more.

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