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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asked Dec 18, 2010 at 10:08
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$\begingroup$ To make your questions better, you might want to include some background or motivation. Why are you interested? What have you tried already? etc. $\endgroup$– Gjergji ZaimiCommented Dec 18, 2010 at 10:43
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$\begingroup$ one can look at mathoverflow.net/questions/49751/… $\endgroup$– Asterios GkantzounisCommented Dec 18, 2010 at 10:47
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$\begingroup$ i changed the question according to the answer that Gjergji gave me $\endgroup$– Asterios GkantzounisCommented Dec 18, 2010 at 10:52
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$\begingroup$ is this allowed? $\endgroup$– Asterios GkantzounisCommented Dec 18, 2010 at 11:02
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$\begingroup$ 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. $\endgroup$– Anton GeraschenkoCommented Dec 22, 2010 at 18:01
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1 Answer
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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.
answered Dec 18, 2010 at 10:18
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$\begingroup$ if i allow p=p'-2^n too? $\endgroup$ Commented Dec 18, 2010 at 10:20
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10$\begingroup$ 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 :) $\endgroup$ Commented Dec 18, 2010 at 10:24
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1$\begingroup$ This last number is a counterexample to being a sum or difference of a prime and a power of 2, by the way. $\endgroup$ Commented Dec 18, 2010 at 10:26
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$\begingroup$ do you have a good answer to this closed question too?mathoverflow.net/questions/49730/twin-primes-etc-closed $\endgroup$ Commented Dec 18, 2010 at 10:29
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$\begingroup$ and a kind of joke :if i allow p=p'+/-2^m+/-2^n? $\endgroup$ Commented Dec 18, 2010 at 10:38