To my knowledge, $$\sin(\pi x)=0\iff x\in\Bbb{Z}$$ and thus $$x=\sin(\pi x)+x\iff x\in\Bbb{Z}$$ . Using Desmos, I plotted two equations: $$f(x)=\Gamma(x+1)$$ $$g(x)=\Gamma((\sin(\pi x)+x)+1)$$ and observed the points at which the curves intersect. I assumed that, since $$x=\sin(\pi x)+x\iff x\in\Bbb{Z}$$ and therefore $$f(x)=f(\sin(\pi x)+x)\iff x\in\Bbb{Z}$$ , all points of intersection would have integer $x$-coordinates. I was wrong.
It appears that the graphs do intersect at all positive integers (the gamma function is undefined at negative integers) but there is one 'extra' solution for positive $x$ near $(0.1912,0.9205)$ as well as seemingly infinite solutions for negative $x$. According to my hypothesis, there should be no solutions for negative $x$. How can these 'extra' solutions be explained?
Thanks in advance!
If you consider $f : x \mapsto 0$. It is clear that $f(x) = f(x+\sin(\pi x)) \Longleftrightarrow x \in \mathbb{Z}$ is false. I mean $f(x) = f(x+\sin(\pi x)) \Longleftarrow x \in \mathbb{Z}$ will be true but not the reciprocal.