Are there infinitely many Mersenne primes?
Primes of the form 2^p − 1. Fifty-two are known. Whether the list ends is unknown.
▸Motivation
Every even perfect number corresponds to one, by Euler, so this and the odd-perfect-number question together decide the structure of perfect numbers entirely.
The Lenstra–Pomerance–Wagstaff heuristic predicts about e^γ·log x / log 2 of them below x, which is infinite — but heuristics about primes have been wrong before, and nothing here is close to a proof. It is equally unknown whether infinitely many 2^p − 1 are composite.
Marin Mersenne, 1644. Fifty-two known as of 2024.
▸Lean API
import Mathlib.NumberTheory.Divisors
namespace Conjectura.NT015
/-- A Mersenne prime is a prime of the form `2^p − 1`. Fifty-two are known; whether
there are infinitely many is open, as is whether infinitely many are composite. -/
def goal : Prop :=
∀ N : ℕ, ∃ p : ℕ, N < p ∧ p.Prime ∧ Nat.Prime (2 ^ p - 1)
end Conjectura.NT015▸Definition1
▸Related work0
▸For your AI
I am proving a theorem in Lean 4 and submitting it to Conjectura.
## Problem NT015 — Are there infinitely many Mersenne primes?
Primes of the form 2^p − 1. Fifty-two are known. Whether the list ends is unknown.
## Environment (fixed — do not assume anything newer)
- Lean toolchain: `leanprover/lean4:v4.33.0-rc1`
- Mathlib: `v4.33.0-rc1`
If a lemma you want does not exist in that Mathlib, prove it inline instead of
importing something newer.
## The exact statement I must prove
```lean
theorem solution : ∀ N : ℕ, ∃ p : ℕ, N < p ∧ p.Prime ∧ Nat.Prime (2 ^ p - 1) := by
sorry
```
## The file I submit
```lean
import Conjectura.Problems.NT015.Statement
namespace Submission
theorem solution : Conjectura.NT015.goal := by
sorry
end Submission
```
## The Lean definitions of every term in this problem
These are the actual definitions your proof will be checked against. Do not
substitute your own version of any of them.
### Prime
def
```lean
p.Prime
```
Defined in Mathlib.
## Rules — submissions violating these are rejected automatically
1. **Do not change the name or type of `solution`.** It must satisfy the
statement above exactly.
2. **Do not redefine or shadow anything from the problem's Statement module.**
Declaring your own `goal`, or redefining a name it depends on, produces a
proof of a *different* statement and is rejected. This is the single most
common rejection.
3. **No `sorry`** anywhere, including in helper lemmas. It surfaces as the
axiom `sorryAx` and is detected transitively through imports.
4. **No `native_decide`** — it surfaces as `Lean.ofReduceBool` and is not
accepted, because it trusts compiled code rather than the kernel.
5. Only these axioms are permitted: `propext`, `Classical.choice`,
`Quot.sound`.
6. Follow Mathlib style: hypotheses left of the colon, explicit types,
`snake_case` theorem names, `UpperCamelCase` types.
## What I want from you
Here is my argument in informal mathematics:
> [PASTE YOUR PROOF SKETCH HERE]
Turn it into Lean 4 that compiles under the environment above and satisfies the
statement exactly. Where you are unsure a lemma exists in this Mathlib version,
say so explicitly rather than guessing a name.Submissions are not open yet
Conjectura is in beta. You can read every statement, every definition and the Lean behind them, and download the exact files the checker uses — but proofs are not being accepted yet.
The reason is a deliberate order of operations. Accepting a proof means running a stranger’s code and standing behind a verdict, and no statement here yet carries a researcher’s name. A machine-checked answer to a question nobody has vouched for is worth very little, so the vouching comes first.
The English write-ups are also switched off during the beta. Nothing on this page is generated by a model.
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