First Hardy–Littlewood conjecture

formal-conjectures · wikipedia · ams-11

Let $P = (m_1, \dots, m_k)$ be a tuple of positive even integers. Let
$\pi_P(n)$ denote the number of primes $p\leq n$ such that $(p, p + m_1, \dots, p + m_k)$
forms an admissible prime constellation. Let $w(q; m_1, \dots, m_k)$ denote the
number of distinct residues of $0, m_1, \dots, m_k$ modulo $q$, and let
$$
C_P = 2 ^ k\prod_{\substack{q\ \text{prime} \\ q\geq 3}}
\frac{1 - \frac{w(q; m_1, \dots, m_k)}{q}}{\left(1 - \frac{1}{q}\right)^{k+1}}.
$$
Then
$$
\pi_P(n)\sim C_P\int_2^n\frac{dt}{\log^{k+1}t}.
$$

Mathematical status
Open: marked research open in google-deepmind/formal-conjectures at revision cd3d8db4634733a748b2380f80f77ba3e4b9dda0 (checked 2026-09-11).

Formal availability
A Prop definition is supplied for the pinned Lean 4.33.1 environment and must be accepted by the deployed verifier before it is available.

Sources and provenance
Upstream reference cited by formal-conjectures (checked 2026-09-11): https://en.wikipedia.org/wiki/First_Hardy%E2%80%93Littlewood_conjecture
Formal statement provenance (Apache-2.0) (checked 2026-09-11): https://github.com/google-deepmind/formal-conjectures/blob/cd3d8db4634733a748b2380f80f77ba3e4b9dda0/FormalConjectures/Wikipedia/HardyLittlewood.lean
Local target: Corpus.WikipediaHardyLittlewood.first_hardy_littlewood_conjecture
Source SHA-256: 4950b97af5af6a73c2c0d32f9f94ea96628c21f29bfcab51f2336e2c768b5466
Lean v4.33.1; Mathlib 0df444a360eaa60ab8c11dca51a86af692955474; policy kernel-replay-v1.
The deployed accepted environment records the actual immutable verifier image.

Research discussions are not verified proofs. A formal target specifies a precise statement; accepting a target does not prove it. Checked results apply to their exact statements and pinned environments.

Public JSON record