# nt.quantity idea – Unique prime factorisation of actual numbers Answer

Hello expensive customer to our community We will proffer you an answer to this query nt.quantity idea – Unique prime factorisation of actual numbers ,and the respond will breathe typical by documented info sources, We welcome you and proffer you fresh questions and solutions, Many customer are questioning concerning the respond to this query.

## nt.quantity idea – Unique prime factorisation of actual numbers

Given the clique of prime numbers $$mathbb{P}$$, it’s identified that the weather of $$log mathbb{P}$$ are dimensionally unbiased $$p_{i neq j}, p_j in mathbb{P}, log p_{i neq j} perp log p_j$$ within the sense that:

$$commence{equation} exists alpha_i in mathbb{Z}, sum_{i=1}^infty alpha_i log p_i = log p_j iff alpha_j = 1 land alpha_{i neq j} = 0 tag{1} aim{equation}$$

Furthermore, if we outline the infinite-dimensional vector area:

$$commence{equation} textual content{span}(log mathbb{P}) = Big{sum_{i=1}^infty alpha_i log p_i < infty lvert p_i in mathbb{P} , alpha_i in mathbb{Z}Big} tag{2} aim{equation}$$

and if we outline $$log mathbb{Q}_+ = {log q |q in mathbb{Q}_+}$$ we might deduce from the exclusive prime factorisation
of the integers that:

$$commence{equation} log mathbb{Q}_+ = textual content{span}(log mathbb{P}) tag{3} aim{equation}$$

Now, it not too long ago occurred to me that utilizing the Riemann rearrangement theorem we might deduce that:

$$commence{equation} forall alpha in mathbb{R} exists q_i in mathbb{Q}, alpha = sum_{i=1}^infty q_i log p_i tag{4} aim{equation}$$

and since $$forall x in mathbb{R}, e^x geq 0$$ we might deduce that each optimistic actual quantity has a exclusive prime factorisation:

$$commence{equation} forall alpha in mathbb{R}_+ exists q_i in mathbb{Q}, alpha = prod_{i=1}^infty p_i^{q_i} tag{5} aim{equation}$$

I cerebrate this gives us with a chic building of the actual numbers when it comes to $$mathbb{P}$$. Might this building breathe identified by a specific designation? (It seems that this explicit building will not be generally taught at universities.)

## Proof of uniqueness:

Let’s outline the partial sums $$alpha_N$$ as follows:

$$commence{equation} alpha = lim_{N to infty} alpha_N = lim_{N to infty} sum_{i=1}^N q_i log p_i tag{6} aim{equation}$$

Now, if we suppose that there exists $$q_i’ in mathbb{Q}$$ such that:

$$commence{equation} alpha_N = prod_{i=1}^N p_i^{q_i} = prod_{i=1}^N p_i^{q_i’} implies forall i, q_i – q_i’ = 0 tag{7} aim{equation}$$

By induction on $$N in mathbb{N}$$, we might resolve that the exponents $$q_i in mathbb{Q}$$ are exclusive.

we are going to proffer you the answer to nt.quantity idea – Unique prime factorisation of actual numbers query through our community which brings all of the solutions from a number of dependable sources.