Hardy–Littlewood conjectures

A family of conjectures predicting asymptotic counts for prime patterns. Their second conjecture estimates prime pairs separated by a fixed even gap using a singular series.

Hardy–Littlewood prime tuples conjecture: A conjecture estimating the number of integers up to a bound for which several fixed linear forms are simultaneously prime. It gives the general framework behind the conjectures’ predictions for prime patterns.
Prime number theorem: The theorem that the number of primes up to x is asymptotic to x divided by its natural logarithm. Its prime-density estimate supplies the baseline for the conjectured pattern counts.
Godfrey Harold Hardy: A British mathematician whose work shaped analytic number theory and mathematical analysis. Hardy co-authored the conjectures with Littlewood and developed methods used in their study.
Prime gaps: Differences between consecutive prime numbers. The prime-pair conjecture predicts the frequency of each fixed even gap.
Bounded gaps between primes: The theorem that some fixed finite bound contains infinitely many differences between consecutive primes. It proves a weaker form of recurring prime proximity than the conjectured fixed-gap asymptotics.
Singular series: An infinite product of local factors that adjusts a predicted density to account for congruence restrictions. It encodes how residue classes modulo each prime affect the frequency of a proposed pattern.
Arithmetic progression: A sequence of integers with a fixed starting value and constant difference. Residue classes modulo primes determine which pattern positions can contain primes.
John Edensor Littlewood: A British mathematician known for major contributions to analysis and number theory. Littlewood co-developed the conjectures and their analytic framework with Hardy.
Prime constellation: A finite pattern of primes at specified offsets that occurs repeatedly among the integers. The conjectures assign predicted asymptotic frequencies to admissible constellations.
Sieve problem: The difficulty of using sieve methods alone to prove that a sparse set contains primes rather than almost-primes. This limitation blocks standard sieve approaches from proving many Hardy–Littlewood predictions.

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