Infinitely many prime numbers of the form $n^{2^k}+1$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T16:56:52Z http://mathoverflow.net/feeds/question/15415 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/15415/infinitely-many-prime-numbers-of-the-form-n2k1 Infinitely many prime numbers of the form $n^{2^k}+1$? Portland 2010-02-16T03:13:29Z 2012-02-25T06:22:33Z <p>I am not a specialist of number theory, so please excuse my ignorance: is the following question still an open problem? Let $k \in \mathbb{N}^*$, are there infinitely many prime numbers of the form $n^{2^k}+1$?</p> http://mathoverflow.net/questions/15415/infinitely-many-prime-numbers-of-the-form-n2k1/15416#15416 Answer by Gerry Myerson for Infinitely many prime numbers of the form $n^{2^k}+1$? Gerry Myerson 2010-02-16T03:16:56Z 2010-02-16T03:16:56Z <p>Yes, it is still an open question</p> http://mathoverflow.net/questions/15415/infinitely-many-prime-numbers-of-the-form-n2k1/15418#15418 Answer by Ben Linowitz for Infinitely many prime numbers of the form $n^{2^k}+1$? Ben Linowitz 2010-02-16T03:34:06Z 2010-02-16T03:47:06Z <p>Your question is still open. It is a special case of <a href="http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H" rel="nofollow">Schinzel's Hypothesis H</a> applied to the polynomial $f(x)=x^{2^k}+1$.</p> <p>As Bjorn mentions in his comment, the case of $k=1$ is a particularly famous unsolved problem. It is the fourth of <a href="http://en.wikipedia.org/wiki/Landau%27s_problems" rel="nofollow">Landau's problems</a> (Edmund Landau was a famous German number theorist during the early twentieth century).</p> http://mathoverflow.net/questions/15415/infinitely-many-prime-numbers-of-the-form-n2k1/15426#15426 Answer by Will Jagy for Infinitely many prime numbers of the form $n^{2^k}+1$? Will Jagy 2010-02-16T05:02:53Z 2010-02-16T05:02:53Z <p>It took me a while to find this: <a href="http://www.pnas.org/content/94/4/1054.full" rel="nofollow">http://www.pnas.org/content/94/4/1054.full</a> </p> <p>Anyway by Friedlander and Iwaniec (1997). They proved that there are infinitely many primes of the form $x^2 + y^4 .$ They mention near the end that they do not have a proof for primes of the form $x^2 + y^6$ but would like one. So there is a way to go to settle $x^2 + 1.$</p> <p>FYI, what I did (not remembering title, authors, anything but the result) was write a program to give the primes $x^2 + y^4$ and put the first dozen in Sloane's sequence site search feature.</p>