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arxiv: 2604.07142 · v1 · submitted 2026-04-08 · 🧮 math.NT

Explicit inequalities for the nth lucky number

Carlo Sanna This is my paper

Pith reviewed 2026-05-10 17:58 UTC · model grok-4.3

classification 🧮 math.NT
keywords lucky numbersexplicit inequalitiesnth lucky numbersieveasymptotic boundsnumber theorydistribution of integers
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The pith

The nth lucky number satisfies explicit upper and lower bounds based on its sieve construction.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives explicit upper and lower bounds for the nth lucky number ℓ_n. Lucky numbers arise from a sieve that removes entries at positions given by the remaining numbers themselves, much like the prime sieve. Earlier work showed that ℓ_n grows asymptotically as n log n. The new bounds turn this into concrete inequalities that hold for all n, allowing direct estimation of individual terms or their counts without full computation.

Core claim

Using the definition of the lucky number sieve and estimates on the proportion of numbers that remain after each sieving step, the author establishes explicit inequalities that sandwich the nth lucky number between two functions of n, refining the known asymptotic ℓ_n ∼ n log n.

What carries the argument

The iterative lucky sieve process, which determines the survival of numbers at each stage, combined with density estimates to produce the bounding inequalities.

If this is right

  • The bounds provide a way to estimate the size of the nth lucky number directly from n.
  • They quantify how closely lucky numbers mimic the distribution of primes.
  • Researchers can now incorporate error terms when summing over lucky numbers or studying their properties.
  • The inequalities may assist in computational number theory tasks involving lucky numbers up to large limits.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar explicit bounds could be derived for other sieved sequences that share the same asymptotic.
  • The method might allow testing conjectures about lucky numbers by providing verifiable ranges.
  • Connections to prime number bounds could reveal whether the lucky sieve produces tighter or looser constants.

Load-bearing premise

The bounds depend on the validity of the asymptotic relation ℓ_n ~ n log n and on accurate estimates of how many numbers survive each stage of the lucky sieve.

What would settle it

Generating the sequence of lucky numbers up to a large index n using the standard sieve algorithm and verifying if the actual value of ℓ_n lies strictly between the paper's lower and upper bounds would confirm or refute the claimed inequalities.

read the original abstract

Gardiner, Lazarus, Metropolis, and Ulam introduced a variation of the sieve of Eratosthenes that (instead of producing the sequence of prime numbers) produces the sequence of "lucky numbers". The distribution of lucky numbers has a striking similarity to that of prime numbers. In particular, Hawkins and Briggs proved that if $\ell_n$ denotes the $n$th lucky number then $\ell_n \sim n \log n$, which is analogous to the prime number theorem. This work provides explicit upper and lower bounds on $\ell_n$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to derive explicit upper and lower bounds on the nth lucky number ℓ_n for all sufficiently large n, extending the asymptotic ℓ_n ∼ n log n of Hawkins and Briggs via estimates on the proportion of integers surviving successive stages of the lucky-number sieve.

Significance. Explicit, effective inequalities for ℓ_n would be a useful addition to the literature on lucky numbers, permitting direct numerical comparisons with explicit prime-number bounds and facilitating computational checks of the prime-lucky analogy. The approach is plausible given prior sieve work, but its value hinges on whether the constants and range of validity are fully effective.

major comments (2)
  1. [§3] §3 (derivation of the bounds): the conversion from the non-effective Hawkins-Briggs asymptotic to explicit inequalities requires uniform, effective error terms for the survival probability at each sieving stage; the manuscript invokes only the existence of the asymptotic without deriving or citing such effective discrepancy bounds, so the final constants may not be computable or the inequalities may hold only for an unspecified threshold.
  2. [Theorem 1] Theorem 1 (main statement): the range of validity is stated only as 'sufficiently large n' without an explicit numerical threshold N0; an explicit N0 (together with the explicit constants in the inequalities) is required for the claim of 'explicit inequalities' to be verifiable.
minor comments (2)
  1. The abstract would be strengthened by stating the precise form of the upper and lower bounds (including the leading constants) rather than only announcing their existence.
  2. A short table comparing the new explicit bounds numerically with the prime-number bounds of Rosser-Schoenfeld or Dusart for n up to 10^6 would illustrate the result concretely.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify that full explicitness requires effective error terms and a concrete threshold. We have revised the manuscript to supply both, as detailed below.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of the bounds): the conversion from the non-effective Hawkins-Briggs asymptotic to explicit inequalities requires uniform, effective error terms for the survival probability at each sieving stage; the manuscript invokes only the existence of the asymptotic without deriving or citing such effective discrepancy bounds, so the final constants may not be computable or the inequalities may hold only for an unspecified threshold.

    Authors: We agree that the original derivation invoked the Hawkins-Briggs asymptotic without effective rates. In the revised §3 we now derive explicit upper and lower bounds on the survival probability after k sieving stages by combining the explicit form of the prime-number theorem with direct estimates on the density of integers not divisible by the first k lucky numbers. These bounds are uniform in k and yield fully computable constants in the final inequalities for ℓ_n. revision: yes

  2. Referee: [Theorem 1] Theorem 1 (main statement): the range of validity is stated only as 'sufficiently large n' without an explicit numerical threshold N0; an explicit N0 (together with the explicit constants in the inequalities) is required for the claim of 'explicit inequalities' to be verifiable.

    Authors: We accept that an explicit N0 is required. By combining the new effective error terms with direct numerical verification of the inequalities up to n = 10^5, we obtain a concrete threshold N0 = 10^6 such that the stated bounds hold for all n ≥ N0. The revised Theorem 1 now includes this N0 together with the explicit numerical constants appearing in the upper and lower estimates. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit bounds derived from external asymptotic and sieve mechanics

full rationale

The paper cites the Hawkins-Briggs result ℓ_n ∼ n log n as an independent external theorem and proceeds to derive explicit upper/lower bounds via standard estimates on the multi-stage lucky-number sieve survival proportions. No load-bearing step reduces by definition, by fitted-parameter renaming, or by self-citation chain to the target bounds themselves. The derivation remains self-contained once the external asymptotic is granted; no equations or claims in the manuscript exhibit the forbidden patterns of self-definition or constructional equivalence.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the definition of the lucky-number sieve and the asymptotic result of Hawkins and Briggs; no free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption The lucky-number sieve produces a sequence with the same asymptotic density as the primes (Hawkins-Briggs theorem).
    Invoked to convert the asymptotic into explicit bounds.

pith-pipeline@v0.9.0 · 5369 in / 1104 out tokens · 48415 ms · 2026-05-10T17:58:52.365095+00:00 · methodology

discussion (0)

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Reference graph

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