explaine answer and steps

- Problem 3. A number is a prime if it cannot be a product of two numbers less than it. For examples, 2,3,5,7,11,13 are prime numbers, but 6,8,10 are not. (1) Prove by contradiction that there are infinite prime numbers. Hint: Assume that there are only finite prime numbers p1,p2,,pk. Think about the number p1p2pk+1 (2) Show that the proof by contradiction can be revised to derive an algorithm to find the (k+1)-th least prime number pk+1 given input of the first k least prime numbers p1,,pk. (3) A twin number is a pair of prime numbers of difference 2. For examples, (3,5), (11,13), and (29,31) are all twin numbers. Search internet about twin number conjecture. Provide its history and current research status