This homework will require you to write proofs. Don't forget that each proof must begin with a statement that you claim to be true. "Claim: ....". After that, we should see the word "Proof followed by a correct and carefully written proof, following the guidelines demonstrated in lecture. 1. [40 pts Give a "constructive" proof for each of the following: a. There exist a, b, c EN (all distinct) such that a b2-c3 b. There exists n N such that n can be written two different ways as the sum of two primes. A prime number is a positive integer, n > 1 that has no positive integer divisors other than 1 and n (Obviously, writing something like 8 5 which is a not a prime. of c 3 and also 8 3 5 does not count!) c. There exists n N such that 3n + 2 is composite. A composite number n is a positive integer n > 1 d. There exist a,b,c N such that a is a factor of be, but a is not a factor of b and a is not a factor