Dickson.lean`

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Annotation provenance. This overlay contains 10 curated AI-written annotations and 7190 mechanical source-derived annotations. They have not been human-reviewed as explanations; treat them as unverified and not mathematical authority. Retained generation history: The 71 original curated annotations were produced in a Claude Sonnet 5.5 session at medium effort; Claude Opus 5.5 is recorded in that session as the advisor model, not the direct author. The bulk source-derived expansion was generated mechanically from pinned source comments by tooling developed in a GPT-5.6 Sol session at ultra reasoning; GPT-5.6 Sol did not originate the quoted source prose. On 2026-10-17 the mechanically generated source-comment annotations of the Formal Conjectures, Carleson, FLT, PFR, CompFiles, Mathematics in Lean and Theorem Proving in Lean overlays were regenerated by the same generator after three corrections made in a Claude Sonnet 5.5 session: titles no longer cut a sentence inside mathematics or end on a dangling word, workflow markers are removed from the quoted prose, and declaration names follow the scanner's namespace resolution. The boilerplate step line was dropped, and comments that were only a row of dashes no longer yield annotations. No new prose was written and no human review is claimed.

Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

5 declarations, 5 contain sorry, 0 declared as axioms

Dickson's conjecture

References:

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/ErdosProblems/891.lean

theorem Dickson.dickson_conjecture Dickson's conjecture If a finite set of linear integer forms sorry line 43 source
Source documentation
Dickson's conjecture If a finite set of linear integer forms $f_i(n) = a_i n+b_i$ satisfies Schinzel condition, there exist infinitely many natural numbers $m$ such that $f_i(m)$ are primes for all $i$.

depends on sorry what this means
Statement uses: Finset, Polynomial, Int, Membership.mem, And, Eq, WithBot, Nat, Polynomial.degree, OfNat.ofNat, BunyakovskyCondition, SchinzelCondition, Infinite, Set.ofPred, Nat.Prime, Int.natAbs, Polynomial.eval
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem dickson_conjecture (fs : Finset ℤ[X]) (hfs : ∀ f ∈ fs, f.degree = 1 ∧ BunyakovskyCondition f)
    (hfs' : SchinzelCondition fs) : Infinite {n : ℕ | ∀ f ∈ fs, (f.eval (n : ℤ)).natAbs.Prime} := by
…
```
Mentions: BunyakovskyCondition, SchinzelCondition
theorem Dickson.generalized_twin_primes The generalized twin-prime conjecture For any positive integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime sorry line 54 source
Source documentation
The generalized twin-prime conjecture For any positive integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, Infinite, Set.ofPred, And, Nat.Prime, HAdd.hAdd, HMul.hMul
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem generalized_twin_primes (k : ℕ) (hk : 0 < k) :
    Infinite {p : ℕ | p.Prime ∧ (p + 2 * k).Prime} := by
…
```
theorem Dickson.infinite_safe_primes The infinitude of Sophie Germain primes There are infinitely many primes $p$ such that $2p + 1$ is prime sorry line 63 source
Source documentation
The infinitude of Sophie Germain primes There are infinitely many primes $p$ such that $2p + 1$ is prime.

depends on sorry what this means
Statement uses: Infinite, Nat, Set.ofPred, And, Prime, HAdd.hAdd, HMul.hMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_safe_primes :
    Infinite {p : ℕ | Prime p ∧ Prime (2 * p + 1)} := by
…
```
theorem Dickson.infinite_cousin_primes The infinitude of cousin primes There are infinitely many primes $p$ such that $p + 4$ is prime sorry line 72 source
Source documentation
The infinitude of cousin primes There are infinitely many primes $p$ such that $p + 4$ is prime.

depends on sorry what this means
Statement uses: Infinite, Nat, Set.ofPred, And, Prime, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_cousin_primes :
    Infinite {p : ℕ | Prime p ∧ Prime (p + 4)} := by
…
```
theorem Dickson.infinite_sexy_primes The infinitude of sexy primes There are infinitely many primes $p$ such that $p + 6$ is prime sorry line 81 source
Source documentation
The infinitude of sexy primes There are infinitely many primes $p$ such that $p + 6$ is prime.

depends on sorry what this means
Statement uses: Infinite, Nat, Set.ofPred, And, Prime, HAdd.hAdd, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_sexy_primes :
    Infinite {p : ℕ | Prime p ∧ Prime (p + 6)} := by
…
```