Determine the values of
47/37 and
3/43
stating any properties of the legendre symbol used.
is there just one method to solve all of these or are there different methods to solve depending on the numbers.
$\displaystyle (47/37) = (10/37)$ ---> Congruence
$\displaystyle =(2/37)(5/37)$ ---> Multiplicative
$\displaystyle (2/37) = -1$ because $\displaystyle 37\equiv \pm 3(\bmod 8)$.
$\displaystyle (5/37) = (37/5)$ ---> Quadradic Reciprocity
$\displaystyle (37/5) = (2/5)$ ---> Congruence
$\displaystyle (2/5) = 1$ because $\displaystyle 5\equiv \pm 1(\bmod 8)$.
Thus, $\displaystyle (10/37) = (2/37)(5/37) = (-1)(1)=-1$.