Goormaghtigh.lean`

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

6 declarations, 1 contain sorry, 0 declared as axioms

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 $10^{700}$

Imports (1)

FormalConjecturesUtil

def Goormaghtigh.repunit The repunit with digits digits in the given base, expressed without division line 38 source
Source documentation
The repunit with digits digits in the given base, expressed without division.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Finset.sum, Finset.range, HPow.hPow
5 instances, coercions and auxiliary constants left out.
```
def repunit (base digits : ℕ) : ℕ :=
…
```
def Goormaghtigh.IsGoormaghtighNumber A number having repunit representations of at least three digits in two distinct bases line 42 source
Source documentation
A number having repunit representations of at least three digits in two distinct bases.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, And, LE.le, OfNat.ofNat, Ne, Eq, Goormaghtigh.repunit
2 instances, coercions and auxiliary constants left out.
```
def IsGoormaghtighNumber (N : ℕ) : Prop :=
…
```
theorem Goormaghtigh.goormaghtigh_conjecture The only Goormaghtigh numbers are $31$ and $8191$ sorry line 50 source
Source documentation
The only Goormaghtigh numbers are $31$ and $8191$.

depends on sorry what this means
Statement uses: Nat, Goormaghtigh.IsGoormaghtighNumber, Or, Eq, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem goormaghtigh_conjecture (N : ℕ) (hN : IsGoormaghtighNumber N) :
    N = 31 ∨ N = 8191 := by
…
```
Mentions: Goormaghtigh.IsGoormaghtighNumber
theorem Goormaghtigh.isGoormaghtighNumber_31 The number $31$ is a repunit in bases $2$ and $5$ line 56 source
Source documentation
The number $31$ is a repunit in bases $2$ and $5$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Goormaghtigh.IsGoormaghtighNumber, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LE.le, Ne, Eq, Goormaghtigh.repunit, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_eq_false, Bool.false and 33 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem isGoormaghtighNumber_31 : IsGoormaghtighNumber 31 := by
…
```
Mentions: Goormaghtigh.IsGoormaghtighNumber
theorem Goormaghtigh.isGoormaghtighNumber_8191 The number $8191$ is a repunit in bases $2$ and $90$ line 62 source
Source documentation
The number $8191$ is a repunit in bases $2$ and $90$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Goormaghtigh.IsGoormaghtighNumber, OfNat.ofNat, Nat
Proof uses: Exists.intro, Exists, And, LE.le, Ne, Eq, Goormaghtigh.repunit, of_eq_true, Eq.trans, True, congr, congrArg, eq_true, Mathlib.Meta.NormNum.isNat_le_true, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_eq_false, Bool.false and 33 more
25 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem isGoormaghtighNumber_8191 : IsGoormaghtighNumber 8191 := by
…
```
Mentions: Goormaghtigh.IsGoormaghtighNumber
theorem Goormaghtigh.repunit_two_digits Allowing two-digit repunits would make $13$ a representation in two distinct bases line 68 source
Source documentation
Allowing two-digit repunits would make $13$ a representation in two distinct bases.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: And, Eq, Nat, Goormaghtigh.repunit, OfNat.ofNat
Proof uses: of_eq_true, Eq.trans, True, congr, congrArg, congrFun', Mathlib.Meta.NormNum.IsNat.to_eq, Finset.sum, Finset.range, HPow.hPow, Eq.rec, Finset.cons, Finset.notMem_range_self, Mathlib.Meta.NormNum.IsNat, HAdd.hAdd, Mathlib.Meta.NormNum.isNat_add, Eq.refl, Mathlib.Meta.NormNum.isNat_pow, Mathlib.Meta.NormNum.isNat_ofNat, Mathlib.Meta.NormNum.IsNat.raw_refl and 18 more
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem repunit_two_digits : repunit 3 3 = 13 ∧ repunit 12 2 = 13 := by
…
```
Mentions: Goormaghtigh.repunit