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)
-
(2) = 1
(3)
,2
(3) = 2
(4)
2
,3
(4) = 2
(5)
,2,3,4
(5) = 4
(6)
2,3,4
,5
(6) = 2
(7)
,2,3,4,5,6
(7) = 6
(8)
2,4,6
,3,5,7
(8) = 4
(9)
3,6
,2,4,5,7,8
(9) = 6
(10)
2,4,6,8,5
,3,7,9
(10) = 4
(11)
,2,3,4,5,6,7,8,9,10
(11) = 10
(12)
2,4,6,8,10,3,9
,5,7,11
(12) = 4
(13)
,2,3,4,5,6,7,8,9,10,11,12,
(13) = 12
(14)
2,4,6,8,10,12,7
,3,5,11,13
(14) = 6
(15)
3,5,9,12,6,10
,2,4,7,8,11,13,14
(15) = 8
(16)
2,4,6,8,10,12,14
,3,5,7,11,13,15
(16) = 8
(17)
,2,3,4,5,6,7,8,9,10,11,12,13,
4,15,16
(17) = 16
(18)
2,3,4,6,8,10,12,14
,5,7,11,13,17,
(18) = 6
(19)
,2,3,4,5,6,7,8,910,11,12,13,14,
5,16,17,18
(19) = 18
(20)
2,4,6,8,10,12,1,4,16,18,5,15
,2,4,5,8,10,11,13,16,17,19
(20) = 8
(21)
3,6,9,12,15,18,7,14
,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
,3,5,7,9,13,15,17,19,21
(22) = 10
(23)
3,6,12,8,15,18,21,16
,2,3,4,5,6,7,8,9,10,11,12,13,14,15
6,17,18,19,20,21
(23) = 22
(24)
2,4,6,8,10,12,14,16,18,20,22
6,3
,5,7,11,13,17,19,23
(24) = 8
Part 1
i. (3) = 2
ii. (8) = 4
iii. (11) = 10
iv. (24) = 8
Trends I spotted:
* from my observations I noticed simple trends like:
(2) = 1
Answer is half the phi number
When you double the phi number the answer for that number is double
the original answer (also these phi numbers are powers of 2)
(4) = 2
(8) = 4
(16) = 8
(32) = 16 (and the trend carries on)
This is the same for when you start the very beginning of the trend with the phi numbers (6) and(10). Their answers are proportional: if you double the phi number the answer is doubled. Through further investigation I made the conclusion that you can't start the 'doubling' trend with any odd phi numbers because their relationship isn't the same as the even phi integers. When you double odd phi numbers they give the same answer to the phi function.
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)
-
(2) = 1
(3)
,2
(3) = 2
(4)
2
,3
(4) = 2
(5)
,2,3,4
(5) = 4
(6)
2,3,4
,5
(6) = 2
(7)
,2,3,4,5,6
(7) = 6
(8)
2,4,6
,3,5,7
(8) = 4
(9)
3,6
,2,4,5,7,8
(9) = 6
(10)
2,4,6,8,5
,3,7,9
(10) = 4
(11)
,2,3,4,5,6,7,8,9,10
(11) = 10
(12)
2,4,6,8,10,3,9
,5,7,11
(12) = 4
(13)
,2,3,4,5,6,7,8,9,10,11,12,
(13) = 12
(14)
2,4,6,8,10,12,7
,3,5,11,13
(14) = 6
(15)
3,5,9,12,6,10
,2,4,7,8,11,13,14
(15) = 8
(16)
2,4,6,8,10,12,14
,3,5,7,11,13,15
(16) = 8
(17)
,2,3,4,5,6,7,8,9,10,11,12,13,
4,15,16
(17) = 16
(18)
2,3,4,6,8,10,12,14
,5,7,11,13,17,
(18) = 6
(19)
,2,3,4,5,6,7,8,910,11,12,13,14,
5,16,17,18
(19) = 18
(20)
2,4,6,8,10,12,1,4,16,18,5,15
,2,4,5,8,10,11,13,16,17,19
(20) = 8
(21)
3,6,9,12,15,18,7,14
,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
,3,5,7,9,13,15,17,19,21
(22) = 10
(23)
3,6,12,8,15,18,21,16
,2,3,4,5,6,7,8,9,10,11,12,13,14,15
6,17,18,19,20,21
(23) = 22
(24)
2,4,6,8,10,12,14,16,18,20,22
6,3
,5,7,11,13,17,19,23
(24) = 8
Part 1
i. (3) = 2
ii. (8) = 4
iii. (11) = 10
iv. (24) = 8
Trends I spotted:
* from my observations I noticed simple trends like:
(2) = 1
Answer is half the phi number
When you double the phi number the answer for that number is double
the original answer (also these phi numbers are powers of 2)
(4) = 2
(8) = 4
(16) = 8
(32) = 16 (and the trend carries on)
This is the same for when you start the very beginning of the trend with the phi numbers (6) and(10). Their answers are proportional: if you double the phi number the answer is doubled. Through further investigation I made the conclusion that you can't start the 'doubling' trend with any odd phi numbers because their relationship isn't the same as the even phi integers. When you double odd phi numbers they give the same answer to the phi function.
