DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

The prime-counting function

Definition

For every real number $x \geq 0$ , the prime-counting function is

π (x) : = # {p prime : p \leq x} .

Remarks

The argument distinguishes this function $π (x)$ from the circle constant $π$ .
The function is right-continuous and changes value exactly at the primes.

Depends on

Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$

Used by

Dependency tree · two levels

15 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

Karl-Dieter Crisman, Number Theory: In Context and Interactive (standard reference, not scraped)
Victor Shoup, A Computational Introduction to Number Theory and Algebra, Version 2 (standard reference, not scraped)