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import FormalConjecturesUtilErdős Problem 1055
Reference: erdosproblems.com/1055
namespace Erdos1055A 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.
def IsOfClass : ℕ+ → ℕ → Prop := fun r ↦
PNat.caseStrongInductionOn (p := fun (_ : ℕ+) ↦ ℕ → Prop) r
(fun p ↦ (p + 1).primeFactors ⊆ {2, 3})
(fun n H p ↦
(∀ (m : ℕ+) (hm : m ≤ n), ¬ H m hm p) ∧
(∀ r ∈ (p + 1).primeFactors,
∃ (m : ℕ+) (hm : m ≤ n), H m hm r) ∧
(∃ r ∈ (p + 1).primeFactors,
∀ (m : ℕ+) (hm : m ≤ n), H m hm r → m = n))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 $r$.
@[category textbook, AMS 11]
theorem exists_p (r : ℕ+) : ∃ p, p.Prime ∧ IsOfClass r p := r:ℕ+⊢ ∃ p, Nat.Prime p ∧ IsOfClass r p
All goals completed! 🐙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 $r$.
noncomputable def p (r : ℕ+) : ℕ :=
open scoped Classical in
Nat.find (exists_p r)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?
@[category research open, AMS 11]
theorem erdos_1055 (r) : {p | p.Prime ∧ IsOfClass r p}.Infinite := r:ℕ+⊢ {p | Nat.Prime p ∧ IsOfClass r p}.Infinite
All goals completed! 🐙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.
@[category research open, AMS 11]
theorem erdos_1055.variants.erdos_limit :
Filter.atTop.Tendsto (fun r ↦ (p r : ℝ) ^ (1 / r : ℝ)) Filter.atTop := ⊢ Filter.Tendsto (fun r ↦ ↑(p r) ^ (1 / ↑↑r)) Filter.atTop Filter.atTop
All goals completed! 🐙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.
@[category research open, AMS 11]
theorem erdos_1055.variants.selfridge_limit :
∃ M, ∀ r, (p r : ℝ) ^ (1 / r : ℝ) ≤ M := ⊢ ∃ M, ∀ (r : ℕ+), ↑(p r) ^ (1 / ↑↑r) ≤ M
All goals completed! 🐙-- TODO(Paul-Lez): formalize the rest of the problems on the page.
end Erdos1055