# Graphing a function.. the definition contains min, max, gcd.

Let $$x=\frac{p}{q}$$, where $$p,q \in \mathbb{N}$$.

Graph the function;

$$f(x)=\left\{\begin{matrix} \frac{\min(p,q)+1}{\max(p,q)+1} & \text{if }\text{GCD}(p,q)=1 \\ & \\ 0 & \text{if }\text{GCD}(p,q)\neq1 \end{matrix}\right.$$

Graphing the function by hand is quite easy to understand, and quite difficult to plot. Unfortunately I do not know any graphing website or software where we can define a function as mentioned. Any help will be appreciated.

I think it is really difficult to plot this function, since it is a very particular and unusual case. However, I have a tip for you. You can generate your own script for plotting it using programming methods. Bellow is an idea of how to build your algorithm.

• define $$p=1$$ and $$q=1$$

• from $$i=1$$ to $$N_1$$ do

• from $$j=1$$ to $$N_2$$ do

• $$p=p+1$$

• $$q=q+1$$

• define $$F=\dfrac{\text{min}(p,q)+1}{\text{max}(p,q)+1}$$

• find common divisors of p and q (this requires a factorization algorithm)

• put their common divisors in decreasing order
• if the highest common divisor is one, write in the external file $$\left(\dfrac{p}{q}, F\right)$$
• else, write in the external file $$\left(\dfrac{p}{q}, 0\right)$$

You can choose the $$N_1$$ and $$N_2$$ values, but remember the bigger $$N_1$$ and $$N_2$$ the higher your program will take to factorize p and q. After that, you can use any simple program to plot your external file.

• If you want to add a pseudocode, the identation will help a lot (like python). So try to make 4 spaces in every new line for identation – Luis Felipe Aug 8 at 14:12
• @LuisFelipe thank you so much. I am a new user here, still learning. Your comment will be of great help for me. – Eduardo Henrique Aug 8 at 14:13
• also, you can select a section and press ALT+K – Luis Felipe Aug 8 at 14:37