The first Hardy Littlewood conjecture, also known as the k-Tuple conjecture is concisely presented here. However, I cannot find a paper explaining how Hardy and Littlewood came to such a conjecture. How is their statement justified? Where can the intuition behind the statement be understood? What paper presents a clear introduction to the conjecture and how it arose?


As pointed out in the comment section, you can read the article "Linear Equations in Primes" by Green and Tao, available on the ArXiv here:


in which they mention the works of Dickson as instrumental in the conception of such conjectures. In particular, you might be interested in reference [12].

In general, the intuition behind such conjectures involving specific sets of prime numbers is to observe their asymptotic behaviour using similar arguments as for all the primes, then use other arguments specific to the set in question to refine that asymptotic. For prime $k$-tuples, we take into account the number of open residue classes relative to the primes in each constellation in order to conjure up a multiplicative constant, similar to the twin-prime constant, that represents these residue classes.

In particular, given a prime constellation with $k$ members denoted by $P_k$ and a positive integer $n$,

$$ \pi_{P_k}(n) \sim C_{P_k} \int _{2}^{n}{dt \over (\log t)^{k}}, $$

where $\pi_{P_k}(n)$ denotes the amount of primes $p\leq n$ such that $(p,p+\ldots)\in P_k$ and $C_{P_k}$ is the constant computed using the various open residue classes relative to the constellation.


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