LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Explicit parameters for the small-gamma logarithmic density

Statement

Let $n$ be an even integer, $L = \log n \geq 1000$ , $γ = 10^{- 6}$ , $p = γ L / n$ , $M = n / 2$ , and $k = ⌈ L / 1000 ⌉$ (the least integer at least $L / 1000$ ). Then $0 < p < 1 / 2, L / 1000 \leq k \leq L / 500 \leq M, M p / k \geq 1 / 4000.$ Moreover, $\log 4000 < 9$ and $9 / 500 + 10^{- 6} < 1 / 50$ .

Facts & Assumptions

[F1]

The ceiling of a real $x$ is the least integer at least $x$ , and belongs to $[x, x + 1)$ (math:lem-ceiling-and-least-positive-integer-threshold).

[F2]

The logarithm is inverse to the exponential (The natural logarithm as the inverse of the exponential function).

[F3]

The exponential is the everywhere convergent series $\exp (x) = \sum_{j \geq 0} x^{j} / j!$ (The real exponential function and the number e by a power series).

[F4]

The exponential is strictly increasing (The exponential function is strictly increasing).

Proof

Given: The objects and hypotheses in the Statement.

Goal: Prove the asserted conclusions.

1.1F1F2F3givenalgebra

Since $L \geq 1000$ , [F1] gives $L / 1000 \leq k < L / 1000 + 1 \leq L / 500$ . The series in [F3] has nonnegative terms at $L > 0$ , so $n = \exp (L) \geq 1 + L > L$ by [F2]. Thus $p < γ < 1 / 2$ and $L / 500 < n / 2 = M$ . Also $M p / k = γ L / (2 k) \geq 250 γ = 1 / 4000$ .

2.1F2F3F4algebra∎

The partial sum of the series for $\exp (9)$ through degree 9 is $10661993 / 2240 > 4000$ ; all further terms are positive. By [F2] and [F4], this proves $\log 4000 < 9$ . Finally $9 / 500 + 10^{- 6} = 18001 / 1000000 < 20000 / 1000000 = 1 / 50$ .

Depends on

Used by · 2

Dependency tree · 2 levels

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