From: rusin@vesuvius.math.niu.edu (Dave Rusin) Subject: Re: polynomials and primes Date: 4 Nov 1999 17:50:54 GMT Newsgroups: sci.math Keywords: polynomials taking prime values; Friedlander and Iwaniec theorem In article <382164CC.93833578@texas.net>, James Buddenhagen wrote: >Helmut Richter wrote: >> >> If I have a polynomial in integer coefficients, how can I know whether >> infinitely many of its values (at integer arguments, of course) are >> primes. There are three cases: >... >[cases deleted] > >I'm hoping someone can refresh my memory on this. First, there is >someone's name associated with the conjecture that, except in 'obvious' >cases all polynomials contain infinitely many primes. I've forgotten >the name. Attributed to Schinzel, I suppose, although it's an obvious conjecture to make. I think Schinzel went a step further and conjectured the density of primes of the given type. >Second, unless I misremember, someone (or some people) recently >announced a proof, via some clever sieving arguments, for some special >rather simple polynomials of degree 4. I was quite surprised when I saw >this, since so far as I know there is still no proof for the polynomial >n^2+1. Perhaps you're remembering the proof announced by Friedlander and Iwaniec that there are infinitely many primes of the form a^2 + b^4 . That's strictly speaking a different class of problem, since it's now a polynomial in two variables, but it's remarkable nonetheless. (Even I can prove there are infinitely many primes of the form a^2+b^2, but numbers of the form a^2 + b^4 are rarer, so it's harder to prove there are infinitely many primes among them.) Details of this result, packaged for the "educated layperson", are at the AMS website: http://www.ams.org/new-in-math/mathnews/prime.html dave ============================================================================== From: gwoegi@fmatbds01.tu-graz.ac.REMOVE.at (Gerhard J. Woeginger) Subject: Re: polynomials and primes Date: Fri, 5 Nov 1999 21:39:44 Newsgroups: sci.math >In <7vi9op$9mo$1@sparcserver.lrz-muenchen.de>, Helmut Richter > said: >. If I have a polynomial in integer coefficients, how can I know whether >. infinitely many of its values (at integer arguments, of course) are >. primes. There are three cases: >. 1) The trivial case: All coefficients have a common prime factor, or the >. polynomial is reducible over the integers. >. 2) The near-trivial case: All values have a common prime factor but it >. is not a common prime factor of the coefficients, e.g. x^3-x+3, whose >. values are all divisible by 3. >. 3) The non-trivial case: anything else. There is an old conjecture by Schinzel and Sierpinski, called hypothesis H: Let P1(x),P2(x),...,Pk(x) be polynomials with integer coefficients and positive leading coefficients. Assume that for every integer d>=2, there exists an integer x such that d does *not* divide the product P1(x)*P2(x)*...*Pk(x). Then there exist infinitely many integers y, such that all values P1(y), P2(y), ..., Pk(y) are simlutaneously prime. The special case with k=1 is your case. The special case with k=2 and P1(x)=x and P2(x)=x+2 is the twin-prime conjecture. To my knowledge, the only special case of hypothesis H that has been proven is Dirichlet's theorem. - Gerhard ___________________________________________________________ Gerhard J. Woeginger (gwoegi@opt.math.tu-graz.ac.at)