The first of the Hardy-Littlewood conjectures. The k-tuple conjecture states that the asymptotic number of prime constellations can be computed explicitly. In particular, unless there is a trivial divisibility condition that stops p, p+a_1, ..., p+a_k from consisting of primes infinitely often, then such prime constellations will occur with an asymptotic density which is computable in terms of a_1, ..., a_k. Let 0<m_1<m_2<...<m_k, then the k-tuple conjecture predicts that the...