Talk:Schinzel's hypothesis H

Hey mathematicians! What about Schinzel's Conjecture about prime gaps?

I have read references to a so-called "Schinzel's Conjecture" that says there is always a prime between x and x+ln(x)^2, whenever x>=8. Is this a consequence of Schinzel's hypothesis H? If so, can this article be updated to state (and explain) this. If not, can the article be updated to point out that there are other so-called Schinzel conjectures besides this one? Thanks! —GraemeMcRae^talk 22:09, 29 December 2007 (UTC)[reply]

It's always possible to edit. If you have a source, go right ahead. I note that you are around periodically, though nobody answered you here. One point is that this does have a specific title. It might be better actually to take Schinzel's name off of it and do a re-direct to this article, simply making it clear at the outset that Hypothesis H is Schinzel's. I see no way to get from this hypothesis to the one you give since this deals with arbitrarily large numbers while yours deals with specific bounds. Definitely No. One reason the article is titled as it now stands may be that the one you give, for instance, is named Schinzel's hypothesis [insert letter]. There may be some other hypotheses of his, like the one you state, that deserve their own entries, and I expect that the article on Schinzel deals with what could be expanded upon by wikipedians.Julzes (talk) 00:27, 21 October 2009 (UTC)[reply]

This is related to Cramér's conjecture (a term sometimes used about variations of what that article says). I have not heard about Schinzel in this context. PrimeHunter (talk) 00:53, 21 October 2009 (UTC)[reply]

Is anybody around who can give Quality Opinions

This article deals with a conjecture that many believe true while many also believe it to be false. This is rather rare in mathematics, with most conjectures either established, refuted, supported or undermined in reasonable time. It seems this is one of those where one can easily waffle, but I'm not an expert. I'm wondering about what experts have said and whether that ought not be placed in the article along with whatever (little?) justification they may give.Julzes (talk) 00:32, 21 October 2009 (UTC)[reply]

Is it really that "controversial"? Peter Sarnak in this recent PDF is working at a mega-conjecture that takes it into higher dimensions. I thought Hypothesis H was really what people expected to be true on standard heuristics. Charles Matthews (talk) 07:09, 21 October 2009 (UTC)[reply]

Oh, I may very well be wrong about this, but what I guess I suggest is that what you said and whatever else is out there should be a part of the article. I didn't use the word 'controversial', though I suppose it's implicit. As the article now so briefly notes, it is an extremely broad proposition. What's "expected on standard heuristics" is not really very good mathematics either, by the way. It's not as though it's been shown equivalent to the Riemann Hypothesis, is it? Why couldn't one deliberately select increasingly worse cases of polynomial sets to the point that hypothesis H is inapplicable is something I'm grappling with. Like I said, I'm a novice. The article needs to be fleshed out as much as is reasonable for this site. It is a pretty significant part of the unsettled bits of mathematics, whether controversial or not, and the reader--like myself--may perhaps want a glimmer more of its status.Julzes (talk) 11:00, 21 October 2009 (UTC)[reply]

Let me try to clarify what I, as a novice, think, since perhaps this will elicit some clarification in a future version of the article. Suppose we do grant that the hypothesis is true up to some number, n, of polynomials--this is definitely not known, but suppose it is. What evidence do we have that there can not be a tipping point for the number of polynomials? It would seem that the full quantitative conjecture referenced in the article would have to be gradually rather than precipitously violated, but what evidence do we have that it is not? If we are really satisfied with the quantitative version for even just one polynomial, perhaps a good reason can be given for an inductive step, but perhaps not. My inclination is that we should probably be pretty satisfied only when the full quantitative version is on solid ground for just two polynomials, since it does not seem like there would be anything special about shifting from n to n+1 for any n>1. Even so, I may be missing some reason that n=3 or some higher number is where we should be satisfied. But for just two polynomials, we can guarantee some conspiracy vis a vis the Chinese Remainder Theorem (I'm thinking), and this may be where there is a problem. If your largish smallest prime factors are consistently forced to come in first only at large values, wouldn't this tend to skew one's expectations. I'm sort of getting a picture that this reasoning is false, but putting why in the article could be of benefit.Julzes (talk) 11:42, 21 October 2009 (UTC)[reply]

Minor change of last sentence

I felt that the last sentence on a global obstruction might better reflect that the hypothesis is so far only conjectural, so I modified the sentence with 'assuming that hypothesis H is in fact correct'. Being what I think is a minor change, I did it prior to any debate. —Preceding unsigned comment added by Julzes (talk • contribs) 21:40, 25 October 2009 (UTC)[reply]

Even broader

It might be helpful to reference Bateman-Horn conjecture, which is a quantitative form of this type of discussion. Charles Matthews (talk) 22:56, 25 October 2009 (UTC)[reply]

Already in the article. That's how I found it.Julzes (talk) 23:15, 25 October 2009 (UTC)[reply]

Introduction needs rewriting

The introduction:

"In mathematics, Schinzel's hypothesis H is a very broad generalisation of conjectures such as the twin prime conjecture. It aims to define the maximum possible scope of a conjecture of the nature that a family f_i(n) of values of irreducible polynomials f(t) should be able to take on prime number values simultaneously, for an integer n, that can be as large as we please. Putting it another way, there should be infinitely many such n, for which each of the f_i(n) are prime numbers."

may be possible to comprehend, but I wouldn't bet on it. It desperately needs rewriting to make it clear.

For example: What do the i's have to do with t ??? What do the f_i(n) have to do with f(t) ??? If f(t) is a family of polynomials in one variable (such as x) that are parametrized by the variable t, shouldn't this be stated explicitly??? Does t run through a family of real numbers or just integers, or else what??? Etc.

And "should be" has nothing to do with the proper grammar for expressing a conjecture. A conjecture is expressed as a declarative statement. Its conjectural status is indicated by how that statement is described when explaining it, not by modifying its internal grammar. (For example: The Twin Prime Conjecture is the assertion: "There exist infinitely many twin primes.")

There is no point in making the introduction so brief that it makes no sense at all. Encyclopedias are not guessing games.Daqu (talk) 20:33, 5 December 2009 (UTC)[reply]

Defend some of these points. Writing f(t) assumes only that t is an indeterminate; not particular opaque. "Should" expresses the heuristic point that there is a restricted set of tuples of polynomials that can be prime at once for many values of n (you can't have t and t + 1, since one will be even, for example). I can expand the lead section, but the intention of these sections is not to give a complete formulation, but to allow the reader to see quickly what the article is about. Charles Matthews (talk) 10:14, 6 December 2009 (UTC)[reply]

Just because you can faintly illuminate an incomprehensible definition on the Talk page doesn't make what's in the article any clearer.

Furthermore, there is no other definition attempted anywhere in the article. Which leaves the article on Schinzel's hypothesis without any section that says what the subject of the article is.

It's really not that hard to say what something is, for anyone who understands it. (That does not include me.)Daqu (talk) 16:59, 10 December 2009 (UTC)[reply]

If you read the section "Formulation of hypothesis H", you will find a formulation. Charles Matthews (talk) 17:23, 12 December 2009 (UTC)[reply]

Assessment comment

The comment(s) below were originally left at Talk:Schinzel's hypothesis H/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

Can this article be updated, please, to discuss the Schinzel conjecture related to prime gaps? See Talk:Schinzel's hypothesis H—GraemeMcRae^talk 22:12, 29 December 2007 (UTC)[reply]

Last edited at 22:13, 29 December 2007 (UTC). Substituted at 02:35, 5 May 2016 (UTC)

Infobox mathematical statement

@Deacon Vorbis:: Can you explain why you are not okay with the infobox here? It packages the logical connections from the "Statement" section (which should really be elsewhere) and the date of origin for the conjecture that's given in "External links". It also really doesn't take away any space from someone who chooses to ignore it. — MarkH21 (talk) 05:28, 24 February 2019 (UTC)[reply]

I've nominated the infobox for deletion; probably best to suspend discussion here unless and until that outcome is a keep. –Deacon Vorbis (carbon • videos) 05:31, 24 February 2019 (UTC)[reply]

Is it "one" or "only one" of the following conditions?

The section Statement begins as follows:

"The hypothesis claims that for every finite collection $\{f_{1},f_{2},\ldots ,f_{k}\}$ of nonconstant irreducible polynomials over the integers with positive leading coefficients, one of the following conditions holds:

# There are infinitely many positive integers $n$ such that all of $f_{1}(n),f_{2}(n),\ldots ,f_{k}(n)$ are simultaneously prime numbers, or

# There is an integer $m>1$ (called a fixed divisor) which always divides the product $f_{1}(n)f_{2}(n)\cdots f_{k}(n)$ . (Or, equivalently: There exists a prime $p$ such that for every $n$ there is an $i$ such that $p$ divides $f_{i}(n)$ .)"