Consider set $T$, a subset of $\mathbb{Q}$, structured as functions in the form of a quotient. Specifically,


Where $N$ and $D$ are functions defined on integers that allow $T$ to be a subset of $\mathbb{Q}$.

The problem is, for chosen values of $n$ and $m$, the elements of $T$ are not always fully reduced fractions. For example, in $T=\left\{\frac{m}{n^2+n+5}\right\}$, the element $\left\{\frac{5}{4^2+4+5}\right\}\in T$ but $\gcd(5,25)\neq 1$

I want split $T$ into a countable union of finite sets defined for only relatively prime numerator and denominator values. For example,


Inorder to do so I must find a way to list the numerator and denominator of simplified fractions in $T$ from the outputs of functions $D$ and $M$.

I determined the denominators of all reduced fractions in $T$ by listing the divisors of all outputs of $D$. This is true for all the simplified elements as long as the numerator and denominator of $T$ do not share common factors, such as $\left\{\frac{m^2}{n^2}\right\}=\left\{\left(\frac{m}{n}\right)^{2}\right\}$.

The list has non-repeated divisors organized from least to greatest. Mathematically I will show this as $\bigcup\limits_{z\in\mathbb{N}}d_{n}^{+}[D(n)]$, where $d_z^{+}[x]$ takes the zth smallest positive and non-repeated divisor of the outputs of $x$.

For example, if $T=\left\{\frac{m}{2^n+n}\right\}$, since the numerator and denominator do not share common factors and $D(n)=2^n+n$, the denominator values of reduced fractions in $T$ is


We get this list by doing the following. Say we take the first positive outputs of $2^n+n$


The divisors of each output are repesented in brackets

$$\left\{\left\{1\right\},\left\{1,3\right\},\left\{1,3,2,6 \right\}, \left\{1,11\right\},\left\{20,1,4,5,10,2\right\},\left\{1,37\right\},\left\{1,70,2,35,7,10\right\}....\right\}$$

We can rearrange these divisors, without repeated values, from least to greatest


The more outputs taken, the closer the list of divisors becomes $\mathbb{N} \ \backslash \left\{0\right\}$


Hence $d_{1}^{+}[2^n+n]=1$, $d_{2}^{+}[2^n+n]=2$, $d_{3}^{+}[2^n+n]=3$ and so on.

Infact, I generalized


When $N(m)$ is linear, I am able to determine the numerator values of simplfied elements in $T$ by listing divisors of the outputs of $N$.


This leads to equations such as





The problem is when $N(m)$ is not linear, I cannot take non-repeated divisors of $N$ to determine the numerator values of reduced elements in $T$.

For example, if we set $T=\left\{\frac{m^2+1}{n^2+n+1}\right\}$, the element $\left\{29/7\right\}\not\in T$; however, 29 is a divisor of an output of $N(m)=m^2+1$. In other words, $d_9[m^2+1]=29$ but $\left\{29/7\right\}\not\in T$

In conclusion, is there a way of determining the numerator of reduced fractions of any set $T$ from the outputs of non-linear $N$?

  • $\begingroup$ What is your concrete problem and functions $D(n),N(m)$ ? $\endgroup$ – reuns Aug 15 '17 at 22:14
  • $\begingroup$ I want to split $T$ into a collection of finite sets with relatively prime numerator and denominator values. The finite sets can be combined into countalby infinite sets in the examples I listed.The functions $D(n)$ and $N(m)$ can be any function that allows $T\subseteq\mathbb{Q}$ $\endgroup$ – Arbuja Aug 15 '17 at 22:18
  • $\begingroup$ In general there is not much more to say. So what is your concrete problem, what are your concrete functions $N(m),D(n)$ ? $\endgroup$ – reuns Aug 15 '17 at 22:25
  • $\begingroup$ Are you still considering $(N(m),D(n)) = (am+b, cn+d)$ and things related ? What is your concrete problem and functions ? $\endgroup$ – reuns Aug 15 '17 at 22:48
  • $\begingroup$ @reuns I looked up concrete problems. I am still unclear what it means mathematically. However, I'll consider $(am^2+bm+c, dn^2+ex+f)$ and $(a^x,b^x)$. $\endgroup$ – Arbuja Aug 16 '17 at 2:46

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