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 defifinitions, rewrite each of the following theorems using quantififiers and predicates. 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 defifinition to indicate that a number is divisible by another. Consider all numbers as positive integers greater than 0. 1. Lagrange’s four-square theorem: Every natural number can be expressed as a sum of four integer squares. 2. 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. 3. Mihăilescu’s theorem: There are no two powers of natural numbers besides 23 and 32 whose values are consecutive.
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 defifinitions, rewrite each of the following theorems using quantififiers and predicates. 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 defifinition to indicate that a number is divisible by another. Consider all numbers as positive integers greater than 0.
1.
Lagrange’s four-square theorem: Every natural number can be expressed
as a sum of four integer squares.
2.
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.
3.
Mihăilescu’s theorem: There are no two powers of natural numbers besides
23 and 32 whose values are consecutive.
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