# The phi-function

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Introduction

MATHS INVESTIGATION THE PHI-FUNCTION Luke Meredith 11 Yellow The Problem With any positive integer (n), the Phi Function of n is ?(n). The Phi number of n is the amount of numbers from 1to n (not including 1 or n) that do no share any common factors with n. If the two or more numbers share no common factor, then the numbers are co-prime. So to put this into practice, the ?(8) = 4. This is because the positive integers less than 8, which have no common factors other than 1 with 8 are 1,3,5,7. This shows 4 of them, which is how the phi number is worked out. Another example is ?(15) = 8. The numbers, which do not have any common factors with 8, from numbers 1 to 8 (excluding 1 and 8), are 1,2,4,6,7,8,11,13,14 = 8 of them. What I am trying to find out is a formula, which will enable me to find the Phi of any number, without going through the painstakingly process of working out the phi for every number. However, you could say that it is not too hard working the phi out for numbers say 1 to a 100. ...read more.

Middle

For the next two, I investigated I decided to take two numbers in each case that multiplied to 12. Example 3. 1) ?(6 x 2) ?12 = 4 2) ?6 x ?2 not same. = 2 x 1 = 2 Example 4 1) ?(4 x 3) ?12 = 4 3) ?4 x ?3 answers are the same. = = 2 x 2 = 4 Once I had done this I realised that there must be a relationship between the two numbers for this equation to work. To carry on my investigation I implied the same tactic to the previous examples, in that both numbers I will choose will give a product (when multiplied) of 30. Example 5 1) ?(5 x 6) ?30 = 8 2) ?5 x ?6 answers are same = = 4 x 2 = 8 Example 6 1) ?(3 x 10) ?30 = 8 2) ?3 x ?10 answers are same = =2 x 4 = 8 Example 7 1) ?(2 x 15) ?30 = 8 2) ?2 x ?15 answers are same = =1 x 8 = 8 In these last three cases the rule ? ...read more.

Conclusion

give a product of the number I had to start with. For example I intend to investigate 36. However to do this I will need two co-prime numbers, which make 36 when multiplied. Which could be 9 and 4. Then I will have to use the formula for p , for both of the numbers. So to not get confused I would use p , for 9 and q , for 4. Finally after all that I would multiply the two remaining numbers together. So I believe that for a general result ?(p q ) is ((p-1)p ) x ((q-1)q ) However for larger numbers I may have to split it up into more than two co-prime numbers. So the formula would then look like this... ((p-1)p ) x ((q-1)q ) x((r-1)r ). Investigating 36. ? (36) = ?(9) x ?(4) ?3 x ? 2 (p-1)p x (q-1)q Investigating 200 Investigating 19600. Conclusion My conclusion is that the rule for p works for everything. However, you need to adapt the formula for larger numbers to ensure that when you use it more than once in the same equation, you don't get confused, lost, or just come out with the wrong answer. ...read more.

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