The Modular Operation x mod m = r denotes that r is the remainder of the division of x by m. For example, 27 mod 4 = 3. If two integers have the same remainder, then they are equivalent.

For example, 27 55 mod 4.

An integer x is called prime if the only two positive integers that evenly divide it are 1 and x.

Using these definitions, rewrite the following theorem using quantifiers and predicates. Note that the theorem is not precisely stated.

You are allowed to use only the predicate Prime (x) that is True if x is a prime, and False otherwise. No other predicates can be used. You can also use either | or the mod definition to indicate that a number is divisible by another. Consider all numbers as positive integers greater than 0.

a) Kaplansky's theorem on quadratic forms (partial): Any prime number p equivalent to 1 mod 16 can be represented by both or neither of the forms x^2 + 32y^2 and x^2 + 64y^2.