Has there been any progress on the Bouniakowsky conjecture? In particular, has anyone been able to prove something for a particular polynomial - or for a class of them?

(I can't seem to find anything, but that could be due to the fact that there seem to be many ways of spelling Bouniakowsky.)

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    $\begingroup$ No. It's still not known for a single polynomial of degree at least two. If it gets proved, you'll hear of it. $\endgroup$ – Felipe Voloch May 30 '12 at 23:41
  • $\begingroup$ @KConrad: that was a silly typo. Thanks. $\endgroup$ – Joshua Seaton May 31 '12 at 0:38

It now goes by the name "Schinzel's Hypothesis H", which has a Wikipedia entry. A quantitative form is known as the "Bateman-Horn Conjecture", which also has a Wikipedia entry.

Short answer: no progress, but better conjectures!

  • $\begingroup$ I figured as much. Thank you, Kevin. $\endgroup$ – Joshua Seaton May 31 '12 at 0:39
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    $\begingroup$ Isn't H much more general than Bouniakowsky? Doesn't it deal with finite collections of polynomials, while Bouniakowsky just deals with a single polynomial? $\endgroup$ – Gerry Myerson May 31 '12 at 7:10
  • $\begingroup$ @gerry: That's right, that's part of the "better" to which I referred. In the single polynomial situation, I think it is common to write "by Schinzel's Hyp. H, ...", while I'd never heard of Bouniakowsky's Conjecture before. $\endgroup$ – Kevin O'Bryant May 31 '12 at 15:03

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