Finding a solutions for an equation

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

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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.

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