• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month
Page
  1. 1
    1
  2. 2
    2
  3. 3
    3
  4. 4
    4
  5. 5
    5
  6. 6
    6
  7. 7
    7
  8. 8
    8
  9. 9
    9
  10. 10
    10
  11. 11
    11
  • Level: GCSE
  • Subject: Maths
  • Word count: 2161

Investigating the Phi function

Extracts from this document...

Introduction

Maths coursework Phi function Investigating the Phi function The phi function is defined for any positive integer(n), as the number of positive integers not greater than and co-prime (have no factor other than 1 in common) to n Example So (12) = 4 because the integers less than 12 which have no factors in common with it except for 1 are 1,5,7,11 i.e. there is 4 of them. I started to investigate the phi function of numbers from 2 to 24 so I could find patterns, which I can use to create a formula for the(n) term (n) Shared factors Not sharing factors (2) - 1 (2) = 1 (3) 1,2 (3) = 2 (4) 2 1,3 (4) = 2 (5) 1,2,3,4 (5) = 4 (6) 2,3,4 1,5 (6) = 2 (7) 1,2,3,4,5,6 (7) = 6 (8) 2,4,6 1,3,5,7 (8) = 4 (9) 3,6 1,2,4,5,7,8 (9) = 6 (10) 2,4,6,8,5 1,3,7,9 (10) = 4 (11) 1,2,3,4,5,6,7,8,9,10 (11) = 10 (12) 2,4,6,8,10,3,9 1,5,7,11 (12) = 4 (13) 1,2,3,4,5,6,7,8,9,10,11,12, (13) = 12 (14) 2,4,6,8,10,12,7 1,3,5,11,13 (14) = 6 (15) 3,5,9,12,6,10 1,2,4,7,8,11,13,14 (15) = 8 (16) 2,4,6,8,10,12,14 1,3,5,7,11,13,15 (16) = 8 (17) 1,2,3,4,5,6,7,8,9,10,11,12,13, 14,15,16 (17) = 16 (18) 2,3,4,6,8,10,12,14 1,5,7,11,13,17, (18) = 6 (19) 1,2,3,4,5,6,7,8,910,11,12,13,14, 15,16,17,18 (19) = 18 (20) 2,4,6,8,10,12,1,4,16,18,5,15 1,2,4,5,8,10,11,13,16,17,19 (20) = 8 (21) 3,6,9,12,15,18,7,14 1,2,4,5,8,10,11,13,16,17,19,20 (21) = 12 (22) 2,4,6,8,10,12,14,16,18,20,11 1,3,5,7,9,13,15,17,19,21 (22) = 10 (23) 3,6,12,8,15,18,21,16 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 16,17,18,19,20,21 (23) ...read more.

Middle

i. (7x4) = (7) x (4) - (28) = 12 - (7) = 6 - (4) = 2 - 12 = 6x2 correct ii. (6x4) = (6) x (4) - (24) = 8 - (6) = 2 - (4) = 2 - 8 = 2x2 wrong The reason why the first one worked and the second didn't could be due to the fact that their numbers are odd , even or a prime number ; and to investigate this I worked out these phi function products and the product of the phi functions of its components: (2x2) =(2)x(2) (4) = 2 (2)=1 2 = 1x1 � (2x3) = (2)x(3) (4)=2 (2)=1 (3)=2 2 = 1x2 (2x4) = (2)x(4) (8) = 4 (2)=1 (4)=2 4= 2x1 � (2x5) = (2)x(5) (10)= 4 (2)=1 (5)=4 4=1x4 (2x6) = (2)x(6) (12)= 4 (2)=1 (6)=2 4= 2x1 � (2x7) = (2)x(7) (14)= 6 (2)=1 (7)=6 (3x2) =(3)x(2) (6)= 2 (3)=2 (2)=1 2 = 1x2 (3x3) = (3)x(3) (9) = 6 (3)=2 6=2x2 � (3x4) = (3)x (4) (12)= 4 (3)=2 (4)=2 4 =2x2 (3x5) = (3)x (5) (15)=8 (3)=2 (5)=4 8 =4x2 (3x6) = (3)x (6) (18)=6 (3)=2 (6)=2 6= 2x2 � (3x7) = (3)x (7) (21)=12 (3)=2 (7)=6 12= 2x6 (4x2) = (4)x (2) (8)=4 (2)=1 (4)=2 4 =2x1 � (4x3) =(4)x (3) (12)=4 (3)=2 (4)=2 4=2x2 (4x4) = (4)x (4) ...read more.

Conclusion

: when p is a prime number you can calculate the final answer by subtracting p from m (m - p). Example (112)=(121) 121-11 =110 the answer I have on my table is (121)=110 ,which proves my method correct. I predict this is the same with any value of p as a positive whole prime number in(p2). (32) =(9) 9 - 3 = 6 correct Example 3 (72)=(49) 49 - 7 = 42 correct so this proves when: p=prime (p2) = (m) m-p = answer * What I also noticed is, to find the answer, for example of (52) you multiply 5 by the number that comes before it which is 4 or by how many sharing factors it has altogether (4) which gives the answer 20. This is correct according to my table. So I predict that (n2) = n x (n - 1), (n is any positive whole integer). Non-prime numbers Examples (42) = 4 x (4 - 1) =12 this is wrong according to my table (142) =14 x (14 - 1) =182 this is wrong according to my table (92) = 9 x (9 -1) = 72 this is wrong according to my table Prime numbers Examples (32) =3 x (3 - 1) =6 this is correct according to my table (72) = 7 x (7 -1) = 42 this is correct according to my table (132) = 13 x (13 - 1) = 156 this is correct according to my table I've noticed that from the examples above that my prediction only works when: P = positive whole prime number ...read more.

The above preview is unformatted text

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Phi Function essays

  1. Identify and explain the rules and equations associated with the Phi function.

    4 5 6 7 8 9 10 11 12=12 ?7=1 2 3 4 5 6=6 ?17=1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16=16 This is because a prime number only has two numbers which divide into it, 1 and itself.

  2. The Phi Function

    Yes 9 1,3,9 Yes 10 1,2,5,10 Yes By looking at the table above, I can tell you that the phi function of 11 is 10 since all integers fit into the expression. (4) ?(24) = 8 The factors of 24 are: 1, 2, 3, 4, 6, 8, 12, and 24.

  1. The totient function.

    of hat answer you can find the Phi value that you desire. The formula is shown below- ?(n) = ?(2 x n) Check: - ?(9)= 6 (My formula) 9 x 2= 18, ?18=6 (proved) 4) As mentioned above a formula also applies for even numbers as well.

  2. Millikan's theory.

    Proper functions are thus eminently relational in that they make possible the interaction between the biological item and its environment. They are also historical (in the evolutionary sense of the term). In her Language, Thought, and Other Biological Categories (henceforth LTOBC), as well as in her introduction to White Queen

  1. The phi function.

    A) Check that: 1- ? (7 x 4) = ? (7) x ? (4) Solution: ? (7 x 4) = ? (7) x ? (4) ? (7 x 4) = ? (28) ? (28) = 27, 25, 23, 19, 17, 15, 13, 11, 9, 5, 3, 1. = ? (28) = 12 ? (7) ?

  2. Describe Aristotle's teachings about the differences between the final cause and the other sorts ...

    Aristotle wrote about the different causes and about how different they are from each other, and without one no other cause can exist. This is another weakness. As it becomes clear that an efficient cause in one thing can also be the formal cause in another, so how I they are so different be the same thing?

  1. The Phi function.

    Even (n) ?(n) 2 1 4 2 6 2 8 4 10 4 12 4 14 6 16 8 18 6 20 8 22 10 24 8 26 12 As we can see that there is no distinguished pattern that can show that the phi values are increasing or decreasing as n increases or decreases.

  2. In this coursework I was asked to investigate the Phi Function (f) of a ...

    (20); 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19; =8 ? (21); 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20; =12 ?

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work