CounterexampleConstruction: 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.

$φ (8) = 4$ but every unit modulo $8$ has square one

Statement refuted

Euler's totient need not equal the exponent of the unit group: $φ (8) = 4$ , but $λ (8) = 2$ .

Facts & Assumptions

Given: The modulus $8$ .

[L1]

$λ (n)$ is the exponent of the unit group (Carmichael's function $λ (n)$ as the exponent of $(Z / n Z)^{\times}$ ).

[L2]

$(Z / 8)^{\times} ≅ C_{2} \times C_{2}$ (For $k \geq 3$ , $(Z / 2^{k} Z)^{\times} ≅ C_{2} \times C_{2^{k - 2}}$ , generated uniquely as $(- 1)^{ε} 5^{j}$ ).

Counterexample

technique · direct

1.1algebra

The four units are $1, 3, 5, 7$ , and each square is congruent to $1$ modulo $8$ .

2.1step 1.1L1L2∎

Thus their group has four elements but exponent $2$ by [L1]; this also agrees with [L2]. Hence $φ (8) = 4 \neq 2 = λ (8)$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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

William Stein, Elementary Number Theory, Example 2.5.10 (standard reference, not scraped)

φ(8)=4φ(8)=4 but every unit modulo 88 has square one