Phi&Sigma Essay, Research Paper

Phi and Sigma

This purpose of this paper is to investigate Euler?s Phi ( ) function. Euler?s phi function, (n), is the number of numbers greater than (n) and relatively to (n).

For Example:

(12)={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12}=[4]

(12)=[4]

The phi of [12] is [4] because there are four divisors between 1-12 whose GCD (greatest common divisor) with [12] is one. To be able to find (n) for larger numbers we must first define the prime factorization of (n) and apply it to the formula where (n) represents the number and (P) represents the prime.

Therefore:

(N)=

f(12)=[4]

4 * 3

2 * 2

=2 *3

Another function used in number theory is the sigma function. For example, (6)=(12) because the factors of 6 are (1, 2, 3, 6). If we add these numbers together we get (1+2+3+6=12). Sigma for large number can be calculated by utilizing the formula below.

(N)= * ?

Example:

(144)

12*12

4*3 4*3

2*2 2*2

=2 *3

* = *

*

*

31*13=403

There are procedures where where (n) is not equal to (m) sentence. Below is a list of values for which sigma; (n)=phi;(m).

The offline program maple V are used to generate:

Sigma;(n)=phi;(m)

4=4

6=6

12=12

8=8

18=18

28=28

24=24

18=18

20=20

42=42

32=32

36=36

24=24

60=60

42=42

40=40

30=30

72=72

48=48

56=56

96=96

44=44

84=84

72=72

54=54

120=120

120=120

80=80

168=168

104=104