Help with this Maths theory problems

1. The Twin Prime Conjecture (unsolved) says that there are infinitely primes p for which p + 2 is also a prime number. In an earlier assignment, you found all primes up to 256. How many pairs of twin primes did you find? 2. Show that there are no prime triplets, i.e., primes p, p + 2, p + 4, other than 3, 5, 7. 3. If n is at least 2 and is not a prime number, then it is called composite. Prove that there are infinitely many positive integers n with n, n + 1, n + 2, n + 3, n + 4 all composite. Can you generalize your proof? 4. Find 3 numbers a, b, c such that (a, b) = (a, c) = (b, c) = 1, but (a, b, c) > 1. Are there infinitely many such triples? Make a conjecture about quadruples. 5. Find the prime factorizations of 10 - 1 and 108 - 1. Use those to compute (106 - 1, 108 - 1). Generalize! 6. Suppose that f(x) = anr" + an-127-1+ . . . + a1x1 + do is a polynomial with integer coefficients. Prove that there is an integer y with f(y) composite. 7. Find a pair of number with god 18 and lom 540. Generalize! 8. Prove that there are infinitely many primes of the form 6k + 5. Generalize! 9. Complete (move to green or, better, blue) the Alcumus topics: "Divisors", "Multiples", "Prime and Composite Basics", "Remainders", "Prime Factorization", "Least Common Multiples", "Greatest Common Divisors", "The Euclidean Algorithm&quot