July 18th, 2018, 05:38 AM  #11  
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  Quote:
It is admittedly difficult to help with proofs because we do not know what definitions, etc. are supposed to be used by the questioner. But In your case, you started with "Every prime number can be expressed as x + (x + 1) or y + (y  1)." That statement is false because 2 is a prime number. Moreover, the true statement that every odd prime can be expressed as the sum of y and y  1 is derived from a more primitive definition that does not apply to prime numbers but to odd numbers. And FINALLY, you say "y, y  1 cannot have common factors because they add up to a prime number". But the truth that y and y  1 cannot have common factors other than 1 is not related to the FALSE proposition that the sum of every integer and its predecessor is a prime, e.g $8 + (8  1) = 16  1 = 15 = 3 * 5.$  
July 18th, 2018, 05:50 AM  #12 
Member Joined: Aug 2015 From: Montenegro (Podgorica) Posts: 37 Thanks: 3 
I meant 8 and (8+1) or 8 and 9 can not have common factors but it is true that this is property of any odd number not just a prime number, because 8 and 7 also don't share common factors and they add up to 15 which is odd and not a prime number.

July 18th, 2018, 06:19 AM  #13  
Senior Member Joined: May 2016 From: USA Posts: 1,148 Thanks: 479  Quote:
So the proof, as mathman indicated in post 2 and as I repeated in my first post, is that all odd numbers can be expressed as specified, all primes are odd except for the smallest, and the number of primes is infinite. It is based on two definitions and one famously proved theorem. As I said, answering questions about proofs is hard because what can be assumed is not known with certainty and because we teach nothing if our suggested proofs are not as careful as we can make them within the constraints of what we have assumed as not requiring proof.  

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