I'm trying to solve the Legendre symbol $\left(\frac{97}{131}\right)$. With my calculation I end up with 1.
(97/131) ={97=1(4)} (131/97)
= (34/97) ={97=1(4)} (97/34)
= (29/34) ={29=1(4)} (34/29)
= (5/29) ={5=1(4)} (29/5) = (4/5) = (2^2/5) = 1
In the solution, it continues at (34/97) like so:
(34/97) = (2/97)(17/97)
= (17/97) ={97=1(4)} (97/17)
= (12/17) = (3/17)(2^2/17) = (3/17)
={17=1(4)} (17/3) = (2/3) ={3=3(8)} -1
So there the solution is -1. I checked both calculations multiple times and they seem fine to me. Did I make a mistake somewhere?
$34=54^2\pmod{131}$ so $97$ is not a quadratic residue of $131$.
I think quadratic reciprocity only works if both are odd primes.