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I need some clarification on (1) Is there any proof to say Mersenne primes $M_p$ are finite or infinite? if there, could you share here.. (2) If Goldbach is conjecture is true, how you can justify the statement" For all finite positive integer 2n can be written as the difference between TWO primes"

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(1) No. (2) What have you tried? – Benjamin Dickman Nov 16 '12 at 6:39
@B.D! for (1) I do not have any idea and I am looking for proof, if there. For (2), I guessed the statement by trail and error. I am not sure, how far I am true. I am looking justification for (2), if possible please share. – vmrfdu123456 Nov 16 '12 at 7:06

Both problems are open. For the first, see the Wikipedia article; amazingly, it is also unknown whether or not there are an infinite number of Mersenne composites. For the second, see this question and the reference to the Prime Pages; however, it is known that every even integer is the difference between a prime and a number which is either a prime or a semi-prime.

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! could you prove the (2). – vmrfdu123456 Nov 16 '12 at 7:59
No, but someday a good mathematician might prove it, or produce a counterexample to refute it. That is what an "open problem" means: nobody knows. – Dan Brumleve Nov 16 '12 at 8:00

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