ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-31

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.

Working out the square-indicator convolution for prime powers

Example

For a prime $p$ ,

(1 * λ) (p^{4}) = 1 - 1 + 1 - 1 + 1 = 1

and

(1 * λ) (p^{3}) = 1 - 1 + 1 - 1 = 0.

So the convolution distinguishes the square prime power $p^{4}$ from the nonsquare prime power $p^{3}$ .

Facts & Assumptions

Given: A prime $p$ .

Verification

technique · direct

1.1

By Liouville's function, $λ (p^{j}) = (- 1)^{j}$ for every $j \geq 0$ . Therefore $(1 * λ) (p^{4}) = \sum_{j = 0}^{4} λ (p^{j}) = \sum_{j = 0}^{4} (- 1)^{j} = 1$ .

givenalgebra

1.2

The same computation gives $(1 * λ) (p^{3}) = \sum_{j = 0}^{3} λ (p^{j}) = \sum_{j = 0}^{3} (- 1)^{j} = 0$ .

givenalgebra

2.1

These explicit values match the square-indicator statement of The convolution $1 * λ$ detects perfect squares for an even and an odd exponent.

step 1.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Tom Sanders, Topics in Analytic Number Theory, Chapter 1 (standard reference, not scraped)