could someone help with this 2 questions plz 23 Prove or di

could someone help with this 2 questions plz? ?

23. Prove or disprove the following conjecture, which is similar to Conjecture i Conjecture: There are infinitely many prime number p for which p+2 and p+4 are also prime numbers ber, then 29. Let n be a positive integer with n 1. Prove that, if n\'+1 is a prime num in the form Ak 1 with k EZ.

Solution

3, 5, and 7 is the only such prime triplet. The proof is easy. Suppose x, x+2, and x+4 are prime and x > 3. Well, x is not a multiple of three because if it were, then x would not be prime. So x is either one more than a multiple of three or two more than a multiple of three. In the first case (x is one more than a multiple of three), x+2 will be a multiple of three and hence won\'t be prime (contrary to our assumption). In the second case, x+4 will be a multiple of three - another contradiction. Thus we have a contradiction in all situations, which means that the assumption must be invalid. Thus 3, 5, 7 is the only such prime triplet.