LemmaStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-27

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.

Complete-quotient tail formula

Statement

Let $α$ be a real number, let $a_{n}$ and $α_{n}$ be its continued-fraction digits and complete quotients, and let $p_{n} / q_{n}$ be the convergents attached to $a_{0}, a_{1}, \dots$ . Whenever $α_{n + 1}$ is defined, $α = \frac{α_{n + 1} p_{n} + p_{n - 1}}{α_{n + 1} q_{n} + q_{n - 1}} .$

Facts & Assumptions

Given: A real number $α$ , its complete quotients $α_{n}$ , and its continued-fraction digits $a_{n}$ .

[F1]

The complete quotients satisfy $α_{n} = a_{n} + 1 / α_{n + 1}$ whenever $α_{n + 1}$ is defined. (Complete quotients in the continued-fraction algorithm).

[F2]

For every $t > 0$ , $[a_{0}; a_{1}, \dots, a_{n}, t] = \frac{t p_{n} + p_{n - 1}}{t q_{n} + q_{n - 1}} .$ (Convergents are given by the standard recurrences and tail formula).

Proof

technique · direct

1.1

Repeatedly substituting the identities of [F1] yields. [given, F1, algebra] $α = [a_{0}; a_{1}, \dots, a_{n}, α_{n + 1}]$ whenever $α_{n + 1}$ is defined.

givenF1algebra

2.1

Since every complete quotient after the first is $> 1$ , in particular $α_{n + 1} > 0$ . [step 1.1, F1, F2] So step 1.1 and [F2] give $α = \frac{α_{n + 1} p_{n} + p_{n - 1}}{α_{n + 1} q_{n} + q_{n - 1}} .$

step 1.1F1F2∎

Depends on

Used by

Dependency tree · two levels

11 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 (standard reference, not scraped)
William Stein, Elementary Number Theory: Primes, Congruences, and Secrets (standard reference, not scraped)