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.

Q: wing statement is true or false. provide a counterexample for each false statement.  x is even iff…

A: In this question, the concept of Modular Arithmetic is applied. Modular Arithmetic Modular…

Q: Apply the Extended Euclidean Algorithm to find the multiplicative inverse of a mod n, provided it…

A:

Q: Assuming Wilson's Theorem, prove that if p prime and (a, p) = 1, then aP-1 = 1 mod p.

A:

Q: Determine which of the following statements are true: a. 54=72(mod16)

A:

Q: Find an inverse modulo 13 of each of the following integers. a) 2 b) 3 c) 5 d) 11

A: if a number n has x (integer) as its inverse modulo 13 then n·x ≡ 1(mod 13) ⇒ n·x = 13k+1 here k is…

Q: Suppose a is a real number. Suppose b additive inverses of a. and care Show that b = c. (Do not use…

A:

Q: Show that there are exactly (p−1)/2 nonzero squares mod p.

A: To prove that there are exactly p-12 nonzero squares mod p.

Q: 3. Find an inverse, if exists, of 12 mod 15.

A:

Q: Consider: for any integers a and b, if (a3 + b3) mod 3 does not equal 0, then (a + b) mod 3 does not…

A: The contrapositive of A implies B is not B implies not A. We use this and solve this problem.…

Q: Show that for any positive integer x, at least one of x, x+ 2, or x+ 4 is divisible by 3

A: The objective is to show that for any positive integer x at one of x, x+ 2, or x+ 4 is divisible by…

Q: The binary expansion 0.abcd... means 23 24 and each binary digit a, b, c, d, ... is either 0 or 1.…

A:

Q: Prove that for any integer x, if x mod 3 does not divide 0, then (x2 −3x + 1) mod 3 = 2.

A:

Q: Let a and b be integers and n a positive integer. Assume also that a and n have a common divisor d…

A: Use the following concepts, to prove the required result. If a divides b then b is a multiple of a.…

Q: Find an integer x satisfying the two congruences: x = 1(mod 3) 2x = 0(mod 6), or show that no such…

A: The given congruences are                                 x≡1 (mod 3)2x≡0 (mod 6)

Statements and Logic

Logic is the analysis of general patterns of thoughts without regard to any specific sense of context.