Number Theory, Part 5

We say that B is a _______ residue modulo m if there is some integer x such that (x^2) == B mod m.

If it is not the case that (x^2) == B mod m is solvable, we say that B is a ________ nonresidue modulo m.

The ______ symbol, (a/p), indicates (via an output of 1 or -1) whether or not a is the type of number mentioned above, modulo p.

Name the theorem: (a/p) == a^((p-1)/2) mod p, where gcd(a, p) = 1, p is prime, and a is an integer, and (a/p) is a symbol (not a fraction).

Name the theorem: (a/p) = (-1)^n, where n is the number of integers from the list a, 2a, 3a, ..., ((p-1)/2)a whose least nonnegative residues modulo p are greater than (p-1)/2.

Name the theorem: (p/q)*(q/p) = (-1)^[((p-1)/2)((q-1)/2))], one with over 100 known proofs.

True or False: (ab/p) = (a/p)*(b/p), where p is prime, and a, b are integers, and (a/p), (b/p) are symbols (not fractions).

True or False: ((a^2)/p) = 1 for all integers a.

True or False: a == b mod p IMPLIES (a/p) = (b/p)