1055.lean`

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

6 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1055

Reference: erdosproblems.com/1055

Imports (1)

FormalConjecturesUtil

def Erdos1055.IsOfClass A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ line 39 source
Source documentation
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor.
The classes partition the primes, so IsOfClass r p says that $p$ is in class exactly $r$: the first conjunct of the successor case rules out every class $\leq r - 1$ for $p$ itself. Without it IsOfClass 2 would also hold of every prime of class $1$, because at $r = 2$ the "with equality" clause quantifies over the single value $m = 1$ and so says nothing.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: PNat, Nat
Proof uses: PNat.caseStrongInductionOn, LE.le, Finset, Nat.primeFactors, HAdd.hAdd, OfNat.ofNat, Insert.insert, Singleton.singleton, And, Not, Membership.mem, Exists, Eq
16 instances, coercions and auxiliary constants left out.
```
def IsOfClass : ℕ+ → ℕ → Prop := fun r ↦
…
```
theorem Erdos1055.exists_p A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ sorry line 54 source
Source documentation
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Show that for each $r$ there exists a prime $p$ of class $r$.

depends on sorry what this means
Statement uses: PNat, Exists, Nat, And, Nat.Prime, Erdos1055.IsOfClass
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 exists_p (r : ℕ+) : ∃ p, p.Prime ∧ IsOfClass r p := by
…
```
Mentions: Erdos1055.p, Erdos1055.IsOfClass
def Erdos1055.p A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ line 62 source
Source documentation
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Let $p_r$ is the least prime in class $r$.

depends on sorry what this means
Statement uses: PNat, Nat
Proof uses: Nat.find, And, Nat.Prime, Erdos1055.IsOfClass, Classical.propDecidable, Erdos1055.exists_p
2 instances, coercions and auxiliary constants left out.
```
noncomputable def p (r : ℕ+) : ℕ :=
…
```
theorem Erdos1055.erdos_1055 A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ sorry line 71 source
Source documentation
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Are there infinitely many primes in each class?

depends on sorry what this means
Statement uses: PNat, Set.Infinite, Nat, Set.ofPred, And, Nat.Prime, Erdos1055.IsOfClass
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 erdos_1055 (r) : {p | p.Prime ∧ IsOfClass r p}.Infinite := by
…
```
Mentions: Erdos1055.p, Erdos1055.IsOfClass
theorem Erdos1055.erdos_1055.variants.erdos_limit A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ sorry line 80 source
Source documentation
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Erdos conjectured that this tends to infinity.

depends on sorry what this means
Statement uses: Filter.Tendsto, PNat, Real, HPow.hPow, Erdos1055.p, HDiv.hDiv, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
35 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1055.variants.erdos_limit :
    Filter.atTop.Tendsto (fun r ↦ (p r : ℝ) ^ (1 / r : ℝ)) Filter.atTop := by
…
```
Mentions: Erdos1055.p
theorem Erdos1055.erdos_1055.variants.selfridge_limit A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$ sorry line 90 source
Source documentation
A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. If $p_r$ is the least prime in class $r$, then how does $p_r^{1/r}$ behave? Selfridge conjectured that this is bounded.

depends on sorry what this means
Statement uses: Exists, Real, PNat, LE.le, HPow.hPow, Erdos1055.p, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, Nat
23 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1055.variants.selfridge_limit :
    ∃ M, ∀ r, (p r : ℝ) ^ (1 / r : ℝ) ≤ M := by
…
```
Mentions: Erdos1055.p