Thanks for all the nice feedback in the CTK Exchange forum.(The symbol = below is the congruent symbol.)
I was reading Hawking's book "God Created the Integers" in which he talks about the quadratic reciprocity theorem.
Two integers p and q are congruent modulo the integer s if and only if (p = q) is evenly divisible by s. Such a congruence is written as p = q (mod s). Using this compact notation, Gauss was able to restate the famous quadratic reciprocity theorem that involves two distinct prime numbers:
If p and q are primes not both congruent to 3 modulo 4, then either:
both x^2 = p (mod q) and x^2 = q (mod p) are solvable or
neither x^2 =p (mod q) nor x^2 =q (mod p) is solvable.
If p and q are primes both congruent to 3 modulo 4, then exactly one of the equations:
x^2 = p (mod q) is x^2 = q (mod p)
QUESTION: what does it really mean when we say "solvable" above?