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

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Graham Pollock 5P                

Introduction

The Phi is described as the number of positive integers less than n (positive integer) which have no factor, other than 1, in common, co-prime, with n.

In my investigation I aim to identify and explain the rules and equations associated with the Phi function.  To go about this I will investigate Φ (n).  This will be further touched upon and will help me investigate and support any conclusions I hope to gain from investigating whether (a x b) = (a) x (b) in certain cases.


Table of Phi’s from 2-40                  

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In my coursework there are some examples which range up to 60 these were obtained from a reliable website and checked over to ensure they were correct.


Investigation

From the Phi table shown previously we can see that some clear patterns emerged.  For example I noticed that for all the prime numbers the Phi is one less than the prime number.

Φ2=1=1                                Φ11=1 2 3 4 5 6 7 8 9 10=10

Φ3=1 2=2                                Φ13=1 2 3 4 5 6 7 8 9 10 11 12=12

Φ7=1 2 3 4 5 6=6                        Φ17=1 2 3 4 5 6 7 ...

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