The Goormaghtigh conjecture

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import FormalConjecturesUtil

/-!
# The Goormaghtigh conjecture

A repunit is a number whose digits in some base are all $1$. Here a nontrivial representation has
at least three digits. The Goormaghtigh conjecture says that $31$ and $8191$ are the only numbers
having nontrivial repunit representations in two different bases.

*References:*
* [Wikipedia](https://en.wikipedia.org/wiki/Goormaghtigh_conjecture)
* J. Grantham,
  [No new Goormaghtigh primes up to $10^{700}$](https://math.colgate.edu/~integers/y98/y98.pdf)
-/

namespace Goormaghtigh

/-- The repunit with `digits` digits in the given base, expressed without division. -/
def repunit (base digits : ℕ) : ℕ :=
  ∑ i ∈ Finset.range digits, base ^ i

/-- A number having repunit representations of at least three digits in two distinct bases. -/
def IsGoormaghtighNumber (N : ℕ) : Prop :=
  ∃ base₁ base₂ digits₁ digits₂ : ℕ,
    2 ≤ base₁ ∧ 2 ≤ base₂ ∧ base₁ ≠ base₂ ∧
      3 ≤ digits₁ ∧ 3 ≤ digits₂ ∧
        repunit base₁ digits₁ = N ∧ repunit base₂ digits₂ = N

/-- The only Goormaghtigh numbers are $31$ and $8191$. -/
@[category research open, AMS 11]
theorem goormaghtigh_conjecture (N : ℕ) (hN : IsGoormaghtighNumber N) :
    N = 31 ∨ N = 8191 := by
  sorry

/-- The number $31$ is a repunit in bases $2$ and $5$. -/
@[category test, AMS 11]
theorem isGoormaghtighNumber_31 : IsGoormaghtighNumber 31 := by
  refine ⟨2, 5, 5, 3, ?_⟩
  norm_num [repunit]

/-- The number $8191$ is a repunit in bases $2$ and $90$. -/
@[category test, AMS 11]
theorem isGoormaghtighNumber_8191 : IsGoormaghtighNumber 8191 := by
  refine ⟨2, 90, 13, 3, ?_⟩
  norm_num [repunit]

/-- Allowing two-digit repunits would make $13$ a representation in two distinct bases. -/
@[category test, AMS 11]
theorem repunit_two_digits : repunit 3 3 = 13 ∧ repunit 12 2 = 13 := by
  norm_num [repunit]

end Goormaghtigh

• Formalization source ↗

On-site verification needs an executable Lean target. None is attached yet.

A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.