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The famous Hardy-Littlewood conjecture on prime-tuples states that if $\{h_1, \cdots, h_k\} = \mathcal{H}$ is an admissible set, that is, for every prime $p$ the set $\mathcal{H}$ does not contain a complete residue system modulo $p$, then there exist infinitely many positive integers $n$ such that the tuple

$$\displaystyle (n+h_1, n+h_2, \cdots, n+h_k)$$

consists of only prime numbers. That is, each entry is prime.

The study of this conjecture is intimately connected to the study of bounded gaps between primes. Indeed, the confirmation that there exist infinitely often bounded gaps between primes is a confirmation of a weaker form of the above conjecture, namely that for fixed positive integer $k$ and a integer $l \geq 1$ (where we would like $l$ to be as small as possible the product

$$\displaystyle (n+h_1)\cdots(n+h_k)$$ contains at most $k+l$ many prime factors (counting multiplicity) between them, and for $l$ sufficiently small this in particular implies that there are at least two factors $n+h_i, n+h_j$ are prime.

My question concerns a higher dimensional analogue of the Hardy-Littlewood conjecture, namely looking at linear forms of more than one variable. Suppose that we are given the linear polynomials $L_j(x_1, \cdots, x_m) = a_1^{(j)} x_1 + \cdots + a_m^{(j)} x_m + h_j$, $1 \leq j \leq k$, such that $a_i^{(j)} \in \mathbb{N}$ for each $i,j$, and the content of each $L_j$ is 1. Then does one expect that for infinitely many $\textbf{x} = (x_1, \cdots, x_m) \in \mathbb{N}^m$ that the tuple

$$\displaystyle (L_1(\textbf{x}), \cdots, L_k(\textbf{x}))$$

is a prime tuple (that is, every entry is prime)?

I suspect that this would be much easier to prove than the Hardy-Littlewood conjecture due to the many degrees of freedom available when $m \geq 2$. If so, has anyone related results already?

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1 Answer 1

up vote 7 down vote accepted

This is sometimes called Dickson's conjecture (questions of this form were raised in his 1904 paper, available here). Tao and Green's paper Linear Equations in the Primes give unconditional results along these lines, when combined with the resolution of the Gowers' norm inverse conjecture by Green, Tao and Ziegler, and the Mobius nilsequence conjecture by Green and Tao.

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