# The Phi Function Investigation

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Introduction

For any positive integer n, the Phi Function o (n) is defined as the number of positive integers less than n which have no factor (other than 1) in common (are co-prime) with n.xml:namespace prefix = o ns = "urn:schemas-microsoft-com:office:office" /> Part one (a) Find the value of: I) o (3) 1, 2, 3 = 2 o (8) 1, 2, 3, 4, 5, 6, 7, 8 = 4 o (11) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 = 10 o (24) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24 = 8 What did I notice? The Phi of both 3 and 11, which are both prime numbers is themselves minus one. So when n is a prime number o n = n - 1 e.g. I would predict that the phi of 17 = 16 o(17) 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 = 16 My formula has proved successful. PART ONE (b) Obtain the Phi function for at least 5 positive integers of your own choice i) o(10) I am using 10 as it is an even number.... 1, 2, 3 ,4 ,5 ,6 , 7, 8 , 9, 10 = 4 ii) ...read more.

Middle

= o (5) x o (10) o(50) = o(5) = 4 x o(10) = 4 probably dun need 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50 = 20 20 = 8 o (4) x o (8) = o ( 4 x 8 ) I am using 4 and 8 as they are two even numbers and one is half of the other... 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32 = 16 o(32) = 16 o(8) = 4 o(4) = 2 4 x 2 = 16 So when I use two even numbers, or when I use a prime and an even number (which has the prime number as it's factor) the equation o (n x m) = o(n) x o(m) doesn't work. Now I will try using 3 sets of a Prime number and a Squared number... o(11) x o(16) = o( 11 x 16 ) o(11) = 10 o(11 x 16) = 176 o(16) ...read more.

Conclusion

There are 9 groups of 2, or I have removed 9 numbers from a total of 27 giving me out an answer of 18. o(27) = 27 - 9 = 18 o(8) = 8 - 4 = 4 I can note a relationship developing, 27 being 3 , and 9 being 3 . Just as 8 being 2 , 4 is 2 . Now I can test this theory on 4 . I can predict that for 4 , because it equals 64, that the function will equal 4 - 4 o64 = 64 - 16 = 48 This equation is wrong, as the o function of 64 is not 48, from this I can conclude that the theory of the relationship only works on prime factors. With this information in mind, I can predict for 5 , that the answer will be 5 - 5 . So for any number (n) to the power of 3 the answer would be n3 - n2. I can now apply this knowledge to predict an equation. For any prime number, to any power. 2 = 16 o16 = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 = 8 16 - 8 = 8 2 - 2 3 = 81 81 - 27 = 3 - 3 In words, the function of any prime number (p) to any power (n) will be: p x p ...read more.

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