FormalConjectures.Wikipedia.Mills

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import FormalConjecturesUtil

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)

namespace Mills

Given any real $A$, IsMills A encodes the statement that $\lfloor A^{3^n}\rfloor,n > 0$ is prime.

abbrev IsMills (A : ℝ) : Prop := ∀ (n : ℕ+), Prime ⌊A ^ (3 ^ (n : ℕ))⌋₊

Mills' theorem (Mills, 1947). There is a real number $A > 1$ such that $\lfloor A^{3^n}\rfloor$ is prime.

@[category research solved, AMS 11]
theorem exists' : ∃ A > 1, IsMills A := by⊢ ∃ A > 1, IsMills A
  sorryAll goals completed! 🐙

For a real $A$, IsMinMills A is the smallest value satisfying IsMills A.

abbrev IsMinMills (A : ℝ) : Prop := IsLeast {x | x > 1 ∧ IsMills x} A

Mills' constant. There is a least Mills number.

@[category research solved, AMS 11]
theorem exists_least : ∃ A, IsMinMills A := by⊢ ∃ A, IsMinMills A
  sorryAll goals completed! 🐙

Mills' constant is irrational (Saito, 2024).

@[category research solved, AMS 11]
theorem irrational {A} (hA : IsMinMills A) :
    Irrational A := byA:ℝhA:IsMinMills A⊢ Irrational A
  sorryAll goals completed! 🐙

Mills' constant lower bound (Caldwell–Cheng, 2005): assuming the Riemann hypothesis, Mills' constant begins at $1.3063778838\ldots$.

@[category research solved, AMS 11]
theorem lower_bound_of_RH (hRH : RiemannHypothesis) {A} (hA : IsMinMills A) :
    A ∈ Set.Ioo (1.3063778838 : ℝ) 1.3063778839 := byhRH:RiemannHypothesisA:ℝhA:IsMinMills A⊢ A ∈ Set.Ioo 1.3063778838 1.3063778839
  sorryAll goals completed! 🐙

end Mills