ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-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.
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$λ (1729) = 36$ although $φ (1729) = 1296$

Example

The number $1729$ satisfies

$λ (1729) = 36 and φ (1729) = 1296.$

Facts & Assumptions

Given: The factorisation $1729 = 7 \cdot 13 \cdot 19$ .

[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]

Euler's totient is multiplicative on coprime positive 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], $λ (1729) = lcm (6, 12, 18) = 36$ .

L1algebra

2.1

By [L2], $φ (1729) = 6 \cdot 12 \cdot 18 = 1296$ , which is strictly larger than the value in step 1.1.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Peter Hackman, Elementary Number Theory, Example C.V.1 (standard reference, not scraped)

λ(1729)=36λ(1729)=36 although φ(1729)=1296φ(1729)=1296