Number of primes < n^3

Conjecture (i): for any integer k > 2, the sequence π(n^k)/n^k (n = 2, 3, …) is strictly decreasing, where π(x) denotes the number of primes not exceeding x. - Zhi-Wei Sun, Oct 17 2015

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-oeis-38098,
  title        = {Number of primes < n^3},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-38098}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

conjecture1. Conjecture (i): for any integer $k > 2$ , the sequence $π (n^{k}) / n^{k}$ ( $n = 2, 3, \dots$ ) is strictly decreasing, where $π (x)$ denotes the number of primes not exceeding $x$ .

Zhi-Wei Sun, Oct 17 2015

conjecture2. Conjecture (ii): all the numbers $π (n^{2}) / n^{2}$ ( $n = 1, 2, 3, \dots$ ) are pairwise distinct. Moreover, we have $π (n^{2}) / n^{2} > π ((n + 1)^{2}) / (n + 1)^{2}$ for all $n > 15646$ .

Zhi-Wei Sun, Oct 17 2015

Number of primes strictly less than $n^{3}$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«38098» (2 statements).

theorem conjecture1 (k : ℕ) (hk : 2 < k) (n : ℕ) (hn : 2 ≤ n) :
    (Nat.primeCounting ((n + 1) ^ k) : ℚ) / ((n + 1) ^ k : ℚ) <
    (Nat.primeCounting (n ^ k) : ℚ) / (n ^ k : ℚ)

theorem conjecture2 :
    (∀ m n : ℕ, 1 ≤ m → 1 ≤ n →
      (Nat.primeCounting (m ^ 2) : ℚ) / (m ^ 2 : ℚ) =
      (Nat.primeCounting (n ^ 2) : ℚ) / (n ^ 2 : ℚ) → m = n) ∧
    (∀ n : ℕ, 15646 < n →
      (Nat.primeCounting ((n + 1) ^ 2) : ℚ) / ((n + 1) ^ 2 : ℚ) <
      (Nat.primeCounting (n ^ 2) : ℚ) / (n ^ 2 : ℚ))

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A038098

Source and licence

Imported from Formal Conjectures (OEIS), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.