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I have arrived to this conjecture in my work, I am not sure that is true or false. So I would appreciate if someone give a counterexample or prove it.

My question: Let n be a non-prime such that n-1 be a prime. Is it true that there exist a prime p such that n-p is a prime as well?

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    $\begingroup$ This is exactly the special case of Goldbach's Conjecture for even numbers that are 1 more than an odd prime. $\endgroup$
    – user22479
    Commented Aug 13, 2012 at 7:27
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    $\begingroup$ Could you give some more details about the work where this question arose? $\endgroup$
    – Yemon Choi
    Commented Aug 13, 2012 at 7:58
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    $\begingroup$ In Goldbach's day, the number 1 counted as a prime, hence the clean conjecture that "every even number is the sum of two (old-fashioned) primes." The modern version, namely that every even number is the sum of two (modern) primes, is stronger exactly by the gap which is the OP's conjecture, hence some natural interest. $\endgroup$
    – David Feldman
    Commented Aug 13, 2012 at 9:33
  • $\begingroup$ retagged as open problem. $\endgroup$
    – Benjamin Steinberg
    Commented Aug 13, 2012 at 14:03
  • $\begingroup$ Deal all thank you so much for you answers. Actually I have arrived to this question in group theory and I have never thought that is an old open problem. I would appreciate, if you let me know any approach concerns this special case. $\endgroup$
    – Tina
    Commented Aug 15, 2012 at 10:22

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The conjecture that any even number (greater than 4) is the sum of two odd primes is well-known to be open. You are asking about a special case that is also open.

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    $\begingroup$ In view of David Feldman's comment, I add that the way this (and these types of question) are approached and seen nowadays this special case seems essentially as hard as the full conjecture. $\endgroup$
    – user9072
    Commented Aug 13, 2012 at 13:07

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