Mersenne.lean`

Explanation legend Lean4 Domain Explanation Language (L4DEL)

L4DEL explanations are a separate layer. They do not change the Lean source. Intuition is explanatory prose; numbered items are a reading path; sources link the wording back to evidence.

L4DEL: Lean4 Domain Explanation Language

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

9 declarations, 4 contain sorry, 0 declared as axioms

Conjectures about Mersenne primes

References:

Imports (1)

FormalConjecturesUtil

def Mersenne.Nat.GivesWagstaffPrime A Wagstaff prime is a prime number of the form $(2^p+1)/3$ line 38 source
Source documentation
A Wagstaff prime is a prime number of the form $(2^p+1)/3$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Odd, Nat.Prime, HDiv.hDiv, HAdd.hAdd, HPow.hPow, OfNat.ofNat
10 instances, coercions and auxiliary constants left out.
```
def GivesWagstaffPrime (p : ℕ) : Prop :=
…
```
def Mersenne.Nat.IsSpecialForm Holds when there is exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$ line 44 source
Source documentation
Holds when there is exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$.

standard axioms: propext what this means
Statement uses: Nat
Proof uses: Exists, Or, Eq, HAdd.hAdd, HPow.hPow, OfNat.ofNat, HSub.hSub
9 instances, coercions and auxiliary constants left out.
```
def IsSpecialForm (p : ℕ) : Prop :=
…
```

def Mersenne.catalanMersenne The Catalan-Mersenne numbers, defined recursively by $c_0 = 2$ and $c_{n+1} = 2^{c_n} - 1$ line 55 source

Source documentation

The Catalan-Mersenne numbers, defined recursively by $c_0 = 2$ and $c_{n+1} = 2^{c_n} - 1$.

standard axioms: propext what this means

Statement uses: Nat

2 instances, coercions and auxiliary constants left out.

def catalanMersenne : ℕ → ℕ
  | 0 => 2
  | n + 1 => 2 ^ catalanMersenne n - 1

/--
A natural number `p` satisfies the statement of the New Mersenne Conjecture if whenever
two of the following conditions hold,
then all three must hold:
1. $2^p-1$ is prime
2. $(2^p+1)/3$ is prime
3. Exists a number `k` such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$
-/

def Mersenne.NewMersenneConjectureStatement A natural number p satisfies the statement of the New Mersenne Conjecture if whenever two of the following conditions hold, then all three must hold: 1 line 67 source
Source documentation
A natural number p satisfies the statement of the New Mersenne Conjecture if whenever two of the following conditions hold, then all three must hold:
1. $2^p-1$ is prime
2. $(2^p+1)/3$ is prime
3. Exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$
standard axioms: propext what this means
Statement uses: Nat
Proof uses: And, Nat.Prime, mersenne, Mersenne.Nat.GivesWagstaffPrime, Mersenne.Nat.IsSpecialForm
```
def NewMersenneConjectureStatement (p : ℕ) : Prop :=
…
```
theorem Mersenne.new_mersenne_conjecture For any odd natural number p if two of the following conditions hold, then all three must hold: 1 sorry line 80 source
Source documentation
For any odd natural number p if two of the following conditions hold, then all three must hold:
1. $2^p-1$ is prime
2. $(2^p+1)/3$ is prime
3. Exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$
depends on sorry what this means
Statement uses: Nat, Odd, Mersenne.NewMersenneConjectureStatement
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem new_mersenne_conjecture (p : ℕ) (hp : Odd p) :
    NewMersenneConjectureStatement p := by
…
```
Mentions: Mersenne.NewMersenneConjectureStatement
theorem Mersenne.new_mersenne_conjecture_of_prime It suffices to check this conjecture for odd primes line 87 source
Source documentation
It suffices to check this conjecture for odd primes. The statement fails at p = 2: mersenne 2 = 3 is prime and 2 = 2 ^ 0 + 1, but 2 gives no Wagstaff prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Nat.Prime, Odd, Mersenne.NewMersenneConjectureStatement
Proof uses: dite, Not, Mersenne.Nat.GivesWagstaffPrime, HDiv.hDiv, HAdd.hAdd, HPow.hPow, OfNat.ofNat, False, Eq, Nat.minFac, Dvd.dvd, Nat.minFac_dvd, Ne, Iff.mpr, Even, Nat.not_even_iff_odd, even_iff_two_dvd, Eq.rec, Nat.Prime.odd_of_ne_two, Nat.minFac_prime and 28 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 40 steps
Open to load the checked before-and-after goal states.
```
theorem new_mersenne_conjecture_of_prime :
    (∀ p, p.Prime → Odd p → NewMersenneConjectureStatement p) →
    ∀ p, Odd p → NewMersenneConjectureStatement p := by
…
```
Mentions: Mersenne.NewMersenneConjectureStatement
theorem Mersenne.new_mersenne_conjecture.variants.prime The New Mersenne Conjecture statement holds for odd primes sorry line 127 source
Source documentation
The New Mersenne Conjecture statement holds for odd primes.

depends on sorry what this means
Statement uses: Nat, Nat.Prime, Odd, Mersenne.NewMersenneConjectureStatement
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem new_mersenne_conjecture.variants.prime (p : ℕ) (hp : p.Prime) (h : Odd p) :
    NewMersenneConjectureStatement p := by
…
```
Mentions: Mersenne.NewMersenneConjectureStatement
theorem Mersenne.infinitely_many_mersenne_primes Are there infinitely many Mersenne primes? sorry line 135 source
Source documentation
Are there infinitely many Mersenne primes?

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

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinitely_many_mersenne_primes :
  answer(sorry) ↔ Set.Infinite { p : ℕ | (mersenne p).Prime } := by
…
```
theorem Mersenne.catalans_mersenne_conjecture The first five Catalan-Mersenne numbers $c_0, \ldots, c_4$ are known to be prime sorry line 145 source
Source documentation
The first five Catalan-Mersenne numbers $c_0, \ldots, c_4$ are known to be prime. Catalan conjectured that they are prime "up to a certain limit". Are all Catalan-Mersenne numbers $c_n$ with $n \geq 5$ prime?

depends on sorry what this means
Statement uses: Iff, True, Nat, GE.ge, OfNat.ofNat, Nat.Prime, Mersenne.catalanMersenne
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem catalans_mersenne_conjecture :
    answer(sorry) ↔ ∀ n ≥ 5, Nat.Prime (catalanMersenne n) := by
…
```
Mentions: Mersenne.catalanMersenne