<s>[INST] Using the Legendre Symbol Which have solutions? $x^{2} \equiv7 \mod{53}$,  $x^{2} \equiv53 \mod{7}$,  $x^{2} \equiv 14 \mod{31}$, $x^{2} \equiv 25\mod{997}$?
I have all of these properties for this Legendre symbol and no idea what to do with it to find whether these have a soltion, let alone solve them if they do. 
 [/INST] They all have solutions. Since $53 \equiv 1 \pmod{4}$, we have
$$\left(\frac{7}{53}\right) = \left(\frac{53}{7}\right) = \left(\frac{4}{7}\right) = \left(\frac{2}{7}\right)^2 = 1,$$
which settles the first two. $31 \equiv 7 \pmod{8}$, so
$$\left(\frac{14}{31}\right) = \left( \frac{2}{31}\right) \left(\frac{7}{31}\right) = \left(\frac{7}{31}\right) = - \left(\frac{31}{7}\right) = - \left(\frac{3}{7}\right) = \left(\frac{7}{3}\right) = \left(\frac13\right) = 1.$$
And finally
$$\left(\frac{25}{997}\right) = \left(\frac{5}{997}\right)^2 = 1.$$
Finding solutions is a different matter, but
$$22^2 = 484 = 9\cdot 53 + 7;\quad 2^2 = 4 \equiv 53 \pmod{7};\quad 13^2 = 169 = 5\cdot 31 + 14; \quad 5^2 = 25.$$
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