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- I. Fermat's test consists of using Fermat's Theorem (an¹=1 (mod n)) with prime n. Just pick any number a E {1, ..., n-1} and check if it satisfies the expression. If not, then it must be a composite number, that is, it is not prime. II. Although Fermat's Theorem is used to check whether a number is prime or not, we can still obtain equality (an¹-1 (mod n)) even when n is not prime. E The numbers for which this situation occurs are called Carmichael numbers or pseudoprimes. III. Both the Miller-Rabin and AKS algorithms (checking a prime number) work for any type of input and are polynomial time. However, only AKS is deterministic. IV. The Miller-Rabin algorithm guarantees when a number is prime, but when it is composite (non-prime) it can only guarantee this characteristic in probabilistic terms. Only the following items are correct: A I, II, III B I, II, IV C I, III, IV D II, III, IV E I, II, III, IVProve that: If n is an even integer, 4 ≤ n ≤ 12, then n is a sum of two prime numbers. Write out all steps of your proof.Prove: For all integers n, if n² is odd, then n is odd. Use a proof by contraposition, as in Lemma 1.1. Let n be an integer. Suppose that n is even, i.e., n = for some integer k. Then n² = is also even.
- 7 Using contraposition, prove that if 3n+4 is even, then n is even, where n is an integer.5. Show that ï-₁(2i − 1) = n² using a proof by induction. =1 6. Show that n 1 using a proof by induction.3. Prove that if n is an integer leaving remainder 1 after division by 4, then = n. Provide the type of your proof and all details.
- Note: For all integers k,n it is true that kn, k+n, and k-n are integers. An integer k is even if and only if there exists an integer r such that k=2r. An integer k is odd if and only if there exists an integer r such that k=2r+1. For every integer k it is true that if k is even then k is not odd. For every integer k it is true that if k is odd then k is not even. For every integer k it is true that if k is not even then k is odd. For every integer k it is true that if k is not odd then k is even. If P then Q means the same thing as P → Q (P implies Q). 3. Consider the argument form pvr .. p→r Is this argument form valid? Prove that your answer is correct. 4. Prove that for every integer d, if d³ is odd then d is odd.1) Direct proofs: A. Prove that for any integer x, the integer x(x + 1) is even. B. Prove that n is even if and only if n^2 is even.) Prove that if n is an integer and 3n + 2 is even, then n is even using a proof by contraposition. a proof by contradication