Mills.lean`

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

6 declarations, 4 contain sorry, 0 declared as axioms

Mills' Theorem

There exists a real $A > 1$ such that $\lfloor A^{3^n}\rfloor$ is prime for every positive integer $n$, where $\lfloor\cdot\rfloor$ denotes the floor function.

The least such $A$ is known as Mills' constant. It is irrational, and assuming the Riemann hypothesis it is approximately $1.3063778838\ldots$.

References:

Wikipedia, Mills' constant
A prime-representing function by W. H. Mills, Bull. Amer. Math. Soc. 53 (1947), 604.
Mills' constant on Wolfram MathWorld.
Mills' constant is irrational by Kota Saito, Mathematika 71 (2025), no. 3, e70027, arXiv:2404.19461.
Determining Mills' Constant .. by Chris K. Caldwell and Yuanyou Cheng, J. Integer Seq. 8 (2005), Article 05.4.1.
OEIS A051021 (decimal expansion of Mills' constant)

Imports (1)

FormalConjecturesUtil

abbrev Mills.IsMills Given any real $A$, IsMills A encodes the statement that $\lfloor A^{3^n}\rfloor,n > 0$ is prime line 49 source
Source documentation
Given any real $A$, IsMills A encodes the statement that $\lfloor A^{3^n}\rfloor,n > 0$ is prime.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: PNat, Prime, Nat, Nat.floor, HPow.hPow, OfNat.ofNat
14 instances, coercions and auxiliary constants left out.
```
abbrev IsMills (A : ℝ) : Prop := ∀ (n : ℕ+), Prime ⌊A ^ (3 ^ (n : ℕ))⌋₊
…
```
theorem Mills.exists' Mills' theorem (Mills, 1947) sorry line 56 source
Source documentation
Mills' theorem (Mills, 1947). There is a real number $A > 1$ such that $\lfloor A^{3^n}\rfloor$ is prime.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Mills.IsMills
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
7 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists' : ∃ A > 1, IsMills A := by
…
```
Mentions: Mills.IsMills
abbrev Mills.IsMinMills For a real $A$, IsMinMills A is the smallest value satisfying IsMills A line 62 source
Source documentation
For a real $A$, IsMinMills A is the smallest value satisfying IsMills A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Real
Proof uses: IsLeast, Set.ofPred, And, GT.gt, OfNat.ofNat, Mills.IsMills
4 instances, coercions and auxiliary constants left out.
```
abbrev IsMinMills (A : ℝ) : Prop := IsLeast {x | x > 1 ∧ IsMills x} A
…
```
Mentions: Mills.IsMills
theorem Mills.exists_least Mills' constant sorry line 69 source
Source documentation
Mills' constant. There is a least Mills number.

depends on sorry what this means
Statement uses: Exists, Real, Mills.IsMinMills
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem exists_least : ∃ A, IsMinMills A := by
…
```
Mentions: Mills.IsMinMills
theorem Mills.irrational Mills' constant is irrational (Saito, 2024) sorry line 76 source
Source documentation
Mills' constant is irrational (Saito, 2024).

depends on sorry what this means
Statement uses: Real, Mills.IsMinMills, Irrational
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem irrational {A} (hA : IsMinMills A) :
    Irrational A := by
…
```
Mentions: Mills.IsMinMills
theorem Mills.lower_bound_of_RH Mills' constant lower bound (Caldwell–Cheng, 2005): assuming the Riemann hypothesis, Mills' constant begins at $1.3063778838\ldots$ sorry line 85 source
Source documentation
Mills' constant lower bound (Caldwell–Cheng, 2005): assuming the Riemann hypothesis, Mills' constant begins at $1.3063778838\ldots$.

depends on sorry what this means
Statement uses: RiemannHypothesis, Real, Mills.IsMinMills, Membership.mem, Set, Set.Ioo, OfScientific.ofScientific, Bool.true
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem lower_bound_of_RH (hRH : RiemannHypothesis) {A} (hA : IsMinMills A) :
    A ∈ Set.Ioo (1.3063778838 : ℝ) 1.3063778839 := by
…
```
Mentions: Mills.IsMinMills