The Modular Operation r mod m = r denotes that r is the remainder = 3. If two integers have the same of the division of x by m. For example, 27 mod 4 remainder, then they are equivalent. For example, 27 = 55 mod 4. An integer r is called prime if the only two positive integers that evenly divide it are 1 and x. Using these definitions, rewrite each of the following theorems using quantifiers and pred- icates. Note that the theorems are 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. Lagrange's four-square theorem: Every natural number can be expressed a. as a sum of four integer squares. b. 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 x2 + 32y? and r2 + 64y2. с. Mihăilescu's theorem: There are no two powers of natural numbers besides 23 and 32 whose values are consecutive.

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