form of primes:prime plus a power of 2? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:46:29Z http://mathoverflow.net/feeds/question/49786 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49786/form-of-primesprime-plus-a-power-of-2 form of primes:prime plus a power of 2? asterios gantzounis 2010-12-18T10:08:00Z 2010-12-18T10:58:51Z <p>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?</p> http://mathoverflow.net/questions/49786/form-of-primesprime-plus-a-power-of-2/49787#49787 Answer by Gjergji Zaimi for form of primes:prime plus a power of 2? Gjergji Zaimi 2010-12-18T10:18:56Z 2010-12-18T10:58:51Z <p>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 <a href="http://oeis.org/A133122" rel="nofollow">"obstinate numbers"</a>. Erdos has proved that there is an infinite arithmetic progression of obstinate numbers.</p> <p>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.</p>