Point 1:
Bruno had a very good answer but I don't think you understand so let me clarify:
To have a number n fit your equation you will need an integer that is both prime and a power of two. The only number that fits this requirement is 2 since 2 is both a prime and a power of two. Any other power of two will not work since they are not prime. Thus there are no twin primes that fit your equation.
For point 2.. I don't fully understand the problem either because I am new to number theory but I will give it a shot and I will gladly take criticism from anyone who understands the problem so I know what I am doing wrong. I will go by what Bruno suggests and work till I find a contradiction. Suppose you have two integers out of the set. Lets use for instance 3n and 4n and set them congruent to each other with some modulo m:
. If you cancel you get
. Then if you work it out algebraically
for some integer k. Well it was stated that
and the only way for that to happen is for k to be a non-integer. We have reached a contradiction and disproved the problem. Can anyone comment on my methodology?
BTW latex in a math forum = freakin awesome.