# Investigating the Phi function

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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.

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