is every prime p equals another prime p' plus or minus a power of 2? p=p'+/-2^n? are there infinitely many primes not of this form?
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To make your questions better, you might want to include some background or motivation. Why are you interested? What have you tried already? etc. – Gjergji Zaimi Dec 18 '10 at 10:43
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one can look at http://mathoverflow.net/questions/49751/chens-theorem-with-congruence-conditions/49782#49782 – Asterios Gkantzounis Dec 18 '10 at 10:47
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i changed the question according to the answer that Gjergji gave me – Asterios Gkantzounis Dec 18 '10 at 10:52
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is this allowed? – Asterios Gkantzounis Dec 18 '10 at 11:02
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It's very much discouraged since it makes the thread look like nonsense. Changing your question to make an existing answer a non-answer is something like inviting to treat somebody to dinner, then slipping out after the meal, sticking them with the bill. – Anton Geraschenko Dec 22 '10 at 18:01
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127 and 331 are counterexamples. It was a conjecture of Polignac that every odd number can be written as a sum of an odd prime and a power of two, but many counterexamples have been found. They are called "obstinate numbers". Erdos has proved that there is an infinite arithmetic progression of obstinate numbers.
Edit (response to the added question): There will be infinitely many such prime counterexamples as a corollary to Erdos' theorem and Dirichlet's theorem on arithmetic progressions. See "Not always buried deep: selections of problems from analytic and combinatorial number theory" by P. Pollack.
Gjergji Zaimi
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10Yes! Apparently a counterexample to that was given by Cohen and Selfridge. 47,867,742,232,066,880,047,611,079 and the proof is left as an exercise :) – Gjergji Zaimi Dec 18 '10 at 10:24
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1This last number is a counterexample to being a sum or difference of a prime and a power of 2, by the way. – Gjergji Zaimi Dec 18 '10 at 10:26
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do you have a good answer to this closed question too?http://mathoverflow.net/questions/49730/twin-primes-etc-closed – Asterios Gkantzounis Dec 18 '10 at 10:29
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@asterious gantzounis: Crocker showed that there are infinitely many odds not of that form, see https://oeis.org/A156695 I don't know if there are infinitely many primes not of that form -- it's been a while since I read his papers -- but I suspect the answer is "yes". – Charles Jul 29 '11 at 21:03
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@GjergjiZaimi where can i find the proof that there exists a $p \neq p' \pm 2^n$ ? – Brad Graham Sep 02 '15 at 00:34