PhiSigma

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]

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

* = *

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

*

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