WieferichPrime.lean`

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7 declarations, 4 contain sorry, 0 declared as axioms

Wieferich primes

A Wieferich prime is a prime $p$ with $2^{p-1} \equiv 1 \pmod{p^2}$. The only known examples are $1093$ and $3511$. It is conjectured that there are infinitely many Wieferich primes; a heuristic argument suggests that the number of Wieferich primes up to $x$ grows like $\log \log x$.

More generally, a prime $p$ is a Wieferich prime to base $a$ if $a^{p-1} \equiv 1 \pmod{p^2}$. Wikipedia's list of unsolved problems also asks whether there are infinitely many Wieferich primes to every base $a > 0$, and whether there is any Wieferich prime to base $47$. It is also not known whether there is any Wieferich prime besides $1093$ and $3511$.

References:

Wikipedia, List of unsolved problems in mathematics
Wikipedia, Wieferich prime
OEIS A001220
P. Ribenboim, Die Welt der Primzahlen, 2nd ed., Springer (2006), pp. 242–243.
PrimeGrid, Wieferich and Wall–Sun–Sun Prime Search

Imports (1)

FormalConjecturesUtil

theorem WieferichPrime.infinite_isWieferichPrime There are infinitely many Wieferich primes sorry line 46 source
Source documentation
There are infinitely many Wieferich primes.

depends on sorry what this means
Statement uses: Set.Infinite, Nat, Set.ofPred, IsWieferichPrime
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_isWieferichPrime : {p : ℕ | IsWieferichPrime p}.Infinite := by
…
```
Mentions: IsWieferichPrime
theorem WieferichPrime.isWieferichPrime_iff Are $1093$ and $3511$ the only Wieferich primes? sorry line 56 source
Source documentation
Are $1093$ and $3511$ the only Wieferich primes? They are the only known ones: PrimeGrid's search, completed in 2022, shows that any other Wieferich prime exceeds $2^{64}$. On the other hand, the heuristic count of $\log \log x$ Wieferich primes up to $x$ predicts that there are infinitely many, see infinite_isWieferichPrime.

depends on sorry what this means
Statement uses: Iff, True, Nat, IsWieferichPrime, 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 isWieferichPrime_iff :
    answer(sorry) ↔ ∀ p, IsWieferichPrime p ↔ p = 1093 ∨ p = 3511 := by
…
```
Mentions: IsWieferichPrime
theorem WieferichPrime.infinite_isWieferichPrimeBase For any given integer $a > 0$, are there infinitely many primes $p$ such that $a^{p-1} \equiv 1 \pmod{p^2}$? sorry line 67 source
Source documentation
For any given integer $a > 0$, are there infinitely many primes $p$ such that $a^{p-1} \equiv 1 \pmod{p^2}$? The case $a = 1$ is trivial, since every prime qualifies (isWieferichPrimeBase_one_iff). So is the case $a = 0$ under our definition (isWieferichPrimeBase_zero_iff), which is why the source's restriction to $a > 0$ is dropped.

depends on sorry what this means
Statement uses: Iff, True, Nat, Set.Infinite, Set.ofPred, IsWieferichPrimeBase
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem infinite_isWieferichPrimeBase :
    answer(sorry) ↔ ∀ a : ℕ, {p : ℕ | IsWieferichPrimeBase a p}.Infinite := by
…
```
Mentions: IsWieferichPrimeBase
theorem WieferichPrime.exists_isWieferichPrimeBase_47 Are there any Wieferich primes to base $47$? sorry line 73 source
Source documentation
Are there any Wieferich primes to base $47$? None is currently known.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, IsWieferichPrimeBase, 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 exists_isWieferichPrimeBase_47 : answer(sorry) ↔ ∃ p, IsWieferichPrimeBase 47 p := by
…
```
Mentions: IsWieferichPrimeBase
theorem WieferichPrime.isWieferichPrime_1093 The prime $1093$ is a Wieferich prime: $2^{1092} \equiv 1 \pmod{1093^2}$ line 78 source
Source documentation
The prime $1093$ is a Wieferich prime: $2^{1092} \equiv 1 \pmod{1093^2}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsWieferichPrime, OfNat.ofNat, Nat
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem isWieferichPrime_1093 : IsWieferichPrime 1093 := by
…
```
Mentions: IsWieferichPrime
theorem WieferichPrime.isWieferichPrime_3511 The prime $3511$ is a Wieferich prime: $2^{3510} \equiv 1 \pmod{3511^2}$ line 83 source
Source documentation
The prime $3511$ is a Wieferich prime: $2^{3510} \equiv 1 \pmod{3511^2}$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: IsWieferichPrime, OfNat.ofNat, Nat
Proof uses: IsWieferichPrimeBase.mk, Mathlib.Meta.NormNum.isNat_prime_2, Mathlib.Meta.NormNum.isNat_ofNat, Eq.refl, Bool, Bool.true, Mathlib.Meta.NormNum.isNat_minFac_4, Mathlib.Meta.NormNum.IsNat.raw_refl, Mathlib.Meta.NormNum.minFacHelper_3, Bool.false, Mathlib.Meta.NormNum.minFacHelper_2, Mathlib.Meta.NormNum.not_prime_mul_of_ble, Mathlib.Meta.NormNum.minFacHelper_0
4 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem isWieferichPrime_3511 : IsWieferichPrime 3511 :=
…
```
Mentions: IsWieferichPrime
theorem WieferichPrime.not_isWieferichPrime_two The prime $2$ is not a Wieferich prime, so no hypothesis excluding it is needed line 88 source
Source documentation
The prime $2$ is not a Wieferich prime, so no hypothesis excluding it is needed.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Not, IsWieferichPrime, OfNat.ofNat, Nat
Proof uses: of_decide_eq_true, id, Eq, Bool, Decidable.decide, Bool.true, Eq.refl
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem not_isWieferichPrime_two : ¬ IsWieferichPrime 2 := by
…
```
Mentions: IsWieferichPrime