MATHmaniac aka CrUNchy: Number Theoretic Function-Part 2

CLICK HERE FOR BLOGGER TEMPLATES AND MYSPACE LAYOUTS �

Number Theoretic Function-Part 2

Labels: Example, Exercises, Theorem | at 5:56 PM

Definition:
An arithmetic function is a function that defined for all positive integers

Definition:
An arithmetic function f is called multiplicative if f(mn) = f(m) f(n) whenever m and n are relatively prime positive integers.It is called multilplicative if
f(mn) = f(m) f(n) for all positive integers m and n.

Let's see...
$\tau (n)$ and $\sigma (n)$ are multiplicative function.
Let say $n=P^{k}$ . Find $\tau (n)$ and $\sigma(n)$ .

$\tau (n)= k + 1$
Positive divisor of n= $P^{0},P^{1},P^{2},...,P^{k}$
$\sigma (n)=P^{0}+P^{1}+P^{2}+...+P^{k}$
by multiplying P,
$P\sigma (n)=P^{1}+P^{2}+P^{3}+...+P^{k+1}$
$P\sigma (n)-\sigma (n) = P^{k+1}-1$
$\sigma (n) (P-1)= P^{k+1}-1$
thus,
$\sigma (n)= (P^{k+1}-1)\div (P-1)$

Get it??
Let's use number,..

$S=2^{0}+2^{1}+2^{2}+...+2^{K}=?$
$2S=2^{1}+2^{2}+2^{3}+...+2^{k+1}$
$2S-S=2^{k+1}- 1$
$S(2-1)=2^{k+1}- 1$
$S=(2^{k+1}- 1)\div (1)$
thus, $S=2^{k+1}- 1$
Let say k=5,so $S=2^{6}- 1=63$

Read Comments

0 comments:

Subscribe to: Post Comments (Atom)