ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.

An index table modulo $17$ turns multiplication into addition modulo $16$

Example

Relative to the primitive root $3$ modulo $17$ , the powers for exponents $0$ through $15$ are

$1, 3, 9, 10, 13, 5, 15, 11, 16, 14, 8, 7, 4, 12, 2, 6.$

Thus, for example, ${ind}_{3} (5) = 5$ , ${ind}_{3} (7) = 11$ , and ${ind}_{3} (6) = 15$ .

Facts & Assumptions

Given: The displayed power table modulo $17$ .

[L1]

The index is the unique exponent class relative to a primitive root (The index ${ind}_{g} (a)$ of a unit relative to a primitive root).

[L2]

Products add indices and powers multiply indices modulo $φ (n)$ (Index calculus: products become sums and powers become scalar multiples modulo $φ (n)$ ).

[L3]

A unit of order $φ (n)$ is a primitive root modulo $n$ (Primitive roots modulo $n$ ).

[L4]

$φ (17) = 16$ ( $φ (1) = 1$ , and $φ (p) = p - 1$ for every prime $p$ ).

Verification

technique · direct

1.1L3L4algebra

Multiplying each displayed residue by $3$ modulo $17$ gives the next one and returns $6$ to $1$ , so the table is correct and contains every nonzero class exactly once. In particular, $3$ has order $16 = φ (17)$ and is primitive by [L3].

2.1step 1.1L1L2L4algebra∎

Since $5 \cdot 7 \equiv 1 (m o d 17)$ , [L2] gives $5 + 11 \equiv 0 (m o d 16)$ for their indices; and $5^{3} \equiv 6 (m o d 17)$ gives $3 \cdot 5 \equiv 15 (m o d 16)$ . Both agree with the entries read using [L1].

Depends on

Used by

Nothing in the library uses this result yet.

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

Peter Hackman, Elementary Number Theory, Example C.I.2 (standard reference, not scraped)

An index table modulo 1717 turns multiplication into addition modulo 1616