Warning: I am an amateur recreational mathematician in highschool who knows everything I know about Fermat's Last Theorem by watching Numberphile videos on YouTube, so please forgive me if I'm getting anything wrong here. 

Fermat's Last Theorem, to my understanding, states that $a^n+b^n \neq c^n$ where $n \gt 2$.  

Could you have $a^n+b^n=c^n$ for $1< n<2$, like a solution(s) for $n=1.2$ or $n=\pi/2$?  

If so, are there any known examples?