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This is a simple question about terminology and provenance.

I just need to sort out the circle of conjectures that generalize and refine the twin prime conjecture.

I've encountered Polignac's conjecture generalizing the twin prime conjecture by replacing pairs $(p,p+2)$ with $(p,p+k)$ for any even $k$. The even more general Hardy-Littlewood conjecture actually predicts the density of any prime constellation not ruled about by residue considerations.

Does the conjecture weaker than Hardy-Littlewood merely predicting the infinite occurrence of such all such constellations enjoy a name unto itself and/or a known first appearance in the literature?

I should also mention that I've encountered the Bunyakovsky conjecture and Schinzel's hypothesis H which generalizes both Bunyakovsky and the (as yet for me) unnamed conjecture of the previous paragraph.

Are there other (canonical) conjectures that belong to this family, either over the integers, or other rings?

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Dickson's conjecture is older and weaker than Schinzel's hypothesis H. primes.utm.edu/glossary/xpage/DicksonsConjecture.html –  Micah Milinovich Jan 21 '11 at 1:26
    
Thanks John! I didn't know about that one. Dickson considers linear polynomials, monic or not. I'm particularly interested in restriction of Dickson to monic polynomials (hence constellations)...does that have it's own name? –  David Feldman Jan 21 '11 at 2:04
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David, you might try to find Dickson's original paper or maybe look in his book "History of the Theory of Numbers, Volume I: Divisibility and Primality." –  Micah Milinovich Jan 21 '11 at 2:53
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2 Answers

up vote 6 down vote accepted

From Green and Tao's "Linear Equations in Primes":

We have been referring to the generalised Hardy-Littlewood conjecture because Hardy and Littlewood [28] in fact only conjectured an asympotic for the number of $n\leq N$ for which the forms $n+b_1,...,n+b_t$ are all prime. If this were generalised to deal with the case of forms $a_1n+b_1,...,a_tn+b_t$ -- the case $d=1$ of Conjecture 1.2 – then a d-parameter version along the lines we have been discussing would follow easily by holding d − 1 of the variables fixed and summing in the remaining one. One has the impression that, had they thought to ask the question, Hardy and Littlewood would easily have produced a conjecture for the asymptotic formula. The name of Dickson is sometimes associated to this circle of ideas. In the 1904 paper [12], he noted the obvious necessary condition on the $a_i$, $b_i$ in order that the forms $a_1n+b_1,...,a_tn+b_t$ might all be prime infinitely often and suggested that this condition might also be sufficient.

The references here are

[12] L. E. Dickson, A new extension of Dirichlet’s theorem on prime numbers, Messenger of Math. 33 (1904), 155–161.

[28] G.H. Hardy and J.E. Littlewood Some problems of “partitio numerorum”; III: On the expression of a number as a sum of primes, Acta Math. 44 (1923), 1–70.

which I'm sure would answer your question, but I don't have access to these at the moment.

EDIT: Further information. The Hardy and Littlewood paper above gives no reference for the conjecture you mention. In Hardy and Wright, after discussing the twin prime conjecture and a similar triplet conjecture, they state

Such conjectures, with larger sets of primes, may be multiplied, but their proof or disproof is at present beyond the resources of mathematics.

I couldn't find any mention of the problem in the Dickson's History.

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Thanks, I think that settles it, unless someone else comes up with the missing link. –  David Feldman Jan 21 '11 at 7:52
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There is also a strengthening of Schinzel's hypothesis H known as the Bateman–Horn conjecture.

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