ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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.

$λ (561) = 80$ and every integer coprime to $561$ has eightieth power congruent to one

Example

One has $λ (561) = 80$ and $φ (561) = 320$ . Consequently every $a$ coprime to $561$ satisfies $a^{80} \equiv 1 (m o d 561)$ .

Facts & Assumptions

Given: The factorisation $561 = 3 \cdot 11 \cdot 17$ .

[L1]

Carmichael's function is the least common multiple of its prime-power values (Carmichael's function on prime powers and its least-common-multiple formula).

[L2]

If $\gcd (a, n) = 1$ , then $a^{λ (n)} \equiv 1 (m o d n)$ (If $\gcd (a, n) = 1$ , then $a^{λ (n)} \equiv 1 (m o d n)$ ).

[L3]

Euler's totient is multiplicative on coprime arguments (Euler's totient is multiplicative: $\gcd (m, n) = 1$ implies $φ (m n) = φ (m) φ (n)$ for positive $m, n$ ).

Verification

technique · direct

1.1

By [L1], $λ (561) = lcm (2, 10, 16) = 80$ , while [L3] gives $φ (561) = 2 \cdot 10 \cdot 16 = 320$ .

L1L3algebra

2.1

Apply [L2] to the value in step 1.1 to obtain $a^{80} \equiv 1 (m o d 561)$ for every $a$ coprime to $561$ .

step 1.1L2∎

Depends on

Used by

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λ(561)=80λ(561)=80 and every integer coprime to 561561 has eightieth power congruent to one