# Confusion with denotations and potentially other things as well

In the article article there is an optimized circuit for Shor's algorithm:

and I'm not sure if $$x=11$$ is the random number $$1 I can choose to calculate $$f(r)= x^r\bmod N$$ for $$r=1,2,3,4\ldots$$ which will lead to the period $$r$$.

Namely, as far as I understand this circuit and the further description which can be found on page 31, $$x$$ is the random number that I can choose. The resulting outcome of the given circuit after running it on the IBM Quantum Composer is $$00100$$ or $$4$$ denoted with $$p$$. Further, in the article, it is said that $$M/p= 8/4 =2$$ where $$M$$ is $$2^n$$, $$n$$ being the number of qubits in the control register giving $$r=2$$.

Inserting $$r=0,1,2,3,4,...$$ in $$x^{r} \bmod 15$$ I obtain $$1,11,1,11,1,\ldots$$ resulting in a period of $$2$$.

Now, what confuses me is that I'm not sure if the result of the given circuit is the period (4) and I got the meaning of $$x$$ wrong, if $$r=2$$ is the period and the result obtained $$(4)$$ isn't, or if both $$p=4$$ and $$r=2$$ are periods of the function, but we have to make sure if there is a smaller one than the obtained one and that's the reason why we divide by $$2$$.

The last paragraph can be summarized as the question: What is the meaning of the obtained result of the given circuit?

The circuit does not return the period itself but a value $$y = M/r$$. It holds that $$M=2^n$$, where $$n$$ is number of qubits used for representation of $$y$$. In your particular case $$n=3$$, hence $$M=8$$. The returned value of $$y$$ is 4 (note that also 0 has high probability but this results is deemed "trivial" and it is neglected). Easily $$r = M/y = 8/4 = 2$$ which is the period.
After that you can calculate factors $$p,q = \gcd\{a^{r/2}\pm1;N\} = \gcd\{11\pm1;15\}$$, i.e. $$p =\gcd\{12;15\}=3$$ and $$q =\gcd\{10;15\}=5$$.