Alphabeta Math
CorollaryStatement: AI-generatedProof: AI-adaptedPipeline-generatedaudited 2026-09-07
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.

Nth prime asymptotic

Statement

If pn is the n-th prime, then pn∼nlog⁡n as n→∞.

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Prime number theorem: As x→∞, π(x)∼x/log⁡x,θ(x)∼x,ψ(x)∼x. These three asymptotic assertions are equivalent.

Proof

1.1

The counting asymptotic implies infinitely many primes, so pn→∞. At these points n=π(pn)=pn(1+o(1))/log⁡pn. Taking logarithms gives log⁡n=log⁡pn−log⁡log⁡pn+o(1), hence log⁡n/log⁡pn→1.

F1
2.1

Rearrange the same positive quantities to obtain pn/(nlog⁡n)=(pn/(nlog⁡pn))(log⁡pn/log⁡n)→1. Each factor is positive for all sufficiently large n, so these limits also yield the usual two-sided epsilon bounds.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources