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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
one can look at… – asterios gantzounis Dec 18 '10 at 10:47
i changed the question according to the answer that Gjergji gave me – asterios gantzounis Dec 18 '10 at 10:52
is this allowed? – asterios gantzounis Dec 18 '10 at 11:02
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
up vote 12 down vote accepted

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.

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if i allow p=p'-2^n too? – asterios gantzounis Dec 18 '10 at 10:20
Yes! 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
This 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
do you have a good answer to this closed question too? – asterios gantzounis Dec 18 '10 at 10:29
and a kind of joke :if i allow p=p'+/-2^m+/-2^n? – asterios gantzounis Dec 18 '10 at 10:38

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