FormalConjectures.Wikipedia.Goormaghtigh

/-
Copyright 2026 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

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

J. Grantham, No new Goormaghtigh primes up to $10^{700}$

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 := byN:ℕhN:IsGoormaghtighNumber N⊢ N = 31 ∨ N = 8191
  sorryAll goals completed! 🐙

The number $31$ is a repunit in bases $2$ and $5$.

@[category test, AMS 11]
theorem isGoormaghtighNumber_31 : IsGoormaghtighNumber 31 := by⊢ IsGoormaghtighNumber 31
  refine ⟨2, 5, 5, 3, ?_⟩⊢ 2 ≤ 2 ∧ 2 ≤ 5 ∧ 2 ≠ 5 ∧ 3 ≤ 5 ∧ 3 ≤ 3 ∧ repunit 2 5 = 31 ∧ repunit 5 3 = 31
  norm_num [repunit]All goals completed! 🐙

The number $8191$ is a repunit in bases $2$ and $90$.

@[category test, AMS 11]
theorem isGoormaghtighNumber_8191 : IsGoormaghtighNumber 8191 := by⊢ IsGoormaghtighNumber 8191
  refine ⟨2, 90, 13, 3, ?_⟩⊢ 2 ≤ 2 ∧ 2 ≤ 90 ∧ 2 ≠ 90 ∧ 3 ≤ 13 ∧ 3 ≤ 3 ∧ repunit 2 13 = 8191 ∧ repunit 90 3 = 8191
  norm_num [repunit]All goals completed! 🐙

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⊢ repunit 3 3 = 13 ∧ repunit 12 2 = 13
  norm_num [repunit]All goals completed! 🐙

end Goormaghtigh