Erdős Problem #409

The question

erdos_409.parts.i. (no description upstream)

erdos_409.parts.i.isTheta. Let $c(n)$ be the minimum number of iterations of $n\mapsto \varphi (n)+1$ before a prime is reached. What is $\Theta (c(n))$?

erdos_409.parts.i.isBigO. Let $c(n)$ be the minimum number of iterations of $n\mapsto \varphi (n)+1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n)=O(g(n))$?

erdos_409.parts.i.isLittleO. Let $c(n)$ be the minimum number of iterations of $n\mapsto \varphi (n)+1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n)=o(g(n))$?

erdos_409.parts.ii. Can infinitely many $n$ reach the same prime under the iteration $n\mapsto \varphi (n)+1$?

erdos_409.parts.iii. What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto \varphi (n)+1$?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«409» (6 statements).

theorem erdos_409.parts.i (n : ℕ) (hn : 0 < n) :
    IsLeast { i | (φ · + 1)^[i] n |>.Prime } erdos_409.parts.i.answer

theorem erdos_409.parts.i.isTheta (c : ℕ → ℕ)
    (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =Θ[atTop] erdos_409.parts.i.isTheta.answer

theorem erdos_409.parts.i.isBigO (c : ℕ → ℕ)
    (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =O[atTop] erdos_409.parts.i.isBigO.answer

theorem erdos_409.parts.i.isLittleO (c : ℕ → ℕ)
    (h : ∀ n > 0, IsLeast { i | (φ · + 1)^[i] n |>.Prime } (c n)) :
    (fun n => (c n : ℝ)) =o[atTop] erdos_409.parts.i.isLittleO.answer

theorem erdos_409.parts.ii :
    ∃ (p : ℕ) (hp : p.Prime), { n | ∃ i, (φ · + 1)^[i] n = p }.Infinite

theorem erdos_409.parts.iii (p : ℕ) (h : p.Prime) :
    { n | ∃ i, (φ · + 1)^[i] n = p }.HasDensity erdos_409.parts.iii.answer

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/409. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

Variants

• erdos_409.variants.sigma
• erdos_409.variants.sigma_termination — If n > 1 then the iteration n↦σ(n) - 1 necessarily reaches a prime.
• erdos_409.variants.sigma_isTheta — Let c(n) be the minimum number of iterations of n↦σ(n) - 1 before a prime is reached. What is Θ(c(n))?
• erdos_409.variants.sigma_isBigO — Let c(n) be the minimum number of iterations of n↦σ(n) - 1 before a prime is reached. Find the simplest function g(n) such that c(n) = O(g(n))?
• erdos_409.variants.sigma_isLittleO — Let c(n) be the minimum number of iterations of n↦σ(n) - 1 before a prime is reached. Find the simplest function g(n) such that c(n) = o(g(n))?
• erdos_409.variants.sigma_prime_termination — Is it true that iterates of n↦σ(n) - 1 always reach a prime?

References

erdosproblems.com/409

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.