## Friday, 28 August 2009

### SPLITTING FTA FUNCTIONS (I)

The Fundamental Theorem of Arithmetic (FTA) grants every natural number, $latex n>1$, a unique factorization of the form:

$latex \displaystyle n=p_1^{\alpha_1}p_2^{\alpha_2}...p_{\omega(n)}^{\alpha_{\omega(n)}} = \prod_{i=1}^{\omega(n)} p_i^{\alpha_i}$

Where $latex \omega(n)$ is the number of distinct prime factors of n.

The arithmetical functions can be evaluated once the factorization of $latex n$ is known, (although there are many of them that can be calculated without factorization)

In fact, the only way to "express some arithmetical property of $latex n$" is that the function, must be dependant on the primes, $latex p_i$ and (or) on the coefficients, $latex \alpha_i$

So the arithmetical functions can be classified, in a psychedelic and unorthodox way of course, in:

1) Functions that depend only on coefficients.

2) Functions that depend only on primes.

3) Functions that depend both on primes and coefficients.

This classification, mathematically speaking, seems to be useless, but it is only an alternative to the alphabetical order, when it comes to deal with this topic.

References:

[1]-D. Joyner, R. Kreminski, J. Turisco @ Applied Abstract Algebra The Fundamental Theorem of Arithmetic
[2]-Arithmetic Function @ Wikipedia Arithmetic Function
[3]-Prime Factor @ Wikipedia Prime Factor