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

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 four square roots of one modulo $128$ are $1, 63, 65, 127$

Example

The solutions of $x^{2} \equiv 1 (m o d 128)$ are

$x \equiv 1, 63, 65, 127 (m o d 128) .$

Facts & Assumptions

Given: The modulus $128 = 2^{7}$ .

[L1]

Every unit modulo $2^{7}$ has a unique form $(- 1)^{ε} 5^{j}$ with $ε \in {0, 1}$ and $j$ modulo $32$ (For $k \geq 3$ , $(Z / 2^{k} Z)^{\times} ≅ C_{2} \times C_{2^{k - 2}}$ , generated uniquely as $(- 1)^{ε} 5^{j}$ ).

Verification

technique · direct

1.1L1algebra

Any solution of $x^{2} = 1$ is a unit, with inverse $x$ . It therefore has the unique form in [L1]. Squaring that form gives $5^{2 j}$ , which is $1$ exactly when $32 ∣ 2 j$ , equivalently $j \equiv 0$ or $16 (m o d 32)$ .

1.2algebra

Repeated squaring gives $5^{16} \equiv 65 (m o d 128)$ .

2.1step 1.1step 1.2L1∎

Combining the two values of $j$ from step 1.1 with the two signs gives $1, - 1, 65, - 65$ , namely $1, 127, 65, 63$ modulo $128$ ; uniqueness in [L1] shows there are no others.

Depends on

For $k\ge3$, $(\mathbb Z/2^k\mathbb Z)^\times\cong C_2\times C_{2^{k-2}}$, generated uniquely as $(-1)^\varepsilon5^j$

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

Peter Hackman, Elementary Number Theory, Example C.IV.9 (standard reference, not scraped)

The four square roots of one modulo 128128 are 1,63,65,1271,63,65,127