Source labels openChecked July 26, 2026

First Hardy–Littlewood conjecture

For integers $x, y \geq 2$ , $\pi(x + y) \leq \pi(x) + \pi(y),$ where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$ .

Number theory AMS 11

Mathematical statement

For integers $x, y \geq 2$ ,

\pi(x + y) \leq \pi(x) + \pi(y),

where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$ .

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

second_hardy_littlewood_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem second_hardy_littlewood_conjecture {x y : ℕ} (hx : 2 ≤ x) (hy : 2 ≤ y) :    SecondHardyLittlewoodConjectureFor x y := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wikipediabackground