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On periodic sequences for algebraic numbers

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Paper breakdownThe authors' claims, conditions and measurements, as JSON Signed receiptProof this review has not been altered
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Thomas Garrity

Number Theory · Submitted 2 Jun 1999

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim Cubic irrationals that are roots of x^3 + k x^2 + x - 1 are shown to be precisely those numbers with purely periodic expansions of period length one.

how it is stated · asserted 1.00 read the record →

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arXiv IDmath/9906016 (version 3)
Pith IDF5E2R7I2, Pith's permanent record of this paper and its review
CitationThomas Garrity. "On periodic sequences for algebraic numbers." arXiv:math/9906016 (1999). https://pith.science/paper/F5E2R7I2
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,143 input tokens.

What the abstract itself states, as probabilities: claims first 0.11 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.40 · compares with prior results 0.80 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots of x^3 + k x^2 + x - 1 are shown to be precisely those numbers with purely periodic expansions of period length one. For general positive integers n, it reduces to a new iteration of an n dimensional simplex.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. contribution 0.99 0.91 0.15 asserted 1.00
s1 The n=2 case is equivalent to the standard continued fraction algorithm. interpretation 0.43 0.27 0.06 asserted 1.00
s2 For n=3, it reduces to a new iteration of the triangle. contribution 0.41 0.78 0.06 asserted 1.00
s3 Cubic irrationals that are roots of x^3 + k x^2 + x - 1 are shown to be precisely those numbers with purely periodic expansions of period length one. result 1.00 0.89 0.41 asserted 1.00
s4 For general positive integers n, it reduces to a new iteration of an n dimensional simplex. method 0.53 0.84 0.16 asserted 1.00

What kind of paper the classifier reports this is: method or tool 0.86 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v3
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to a new iteration of the triangle. Cubic irrationals that are roots of x^3 + k x^2 + x - 1 are shown to be precisely those numbers with purely periodic expansions of period length one. For general positive integers n, it reduces to a new iteration of an n dimensional simplex.

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