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Record W1975398209 · doi:10.1112/s0010437x09004254

Sums of smooth squares

2009· article· en· W1975398209 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.
fundA Canadian funder is recorded on the work.

Bibliographic record

VenueCompositio Mathematica · 2009
Typearticle
Languageen
FieldMathematics
TopicAnalytic Number Theory Research
Canadian institutionsUniversity of Toronto
FundersNatural Sciences and Engineering Research Council of Canada
KeywordsMathematicsInteger (computer science)CombinatoricsExplained sum of squaresLack-of-fit sum of squaresOrder (exchange)Least-squares function approximationNatural numberFermat's theorem on sums of two squaresDiscrete mathematicsNon-linear least squaresStatistics

Abstract

fetched live from OpenAlex

Abstract Let R ( n , θ ) denote the number of representations of the natural number n as the sum of four squares, each composed only with primes not exceeding n θ /2 . When θ > e −1/3 a lower bound for R ( n , θ ) of the expected order of magnitude is established, and when θ >365/592, it is shown that R ( n , θ )>0 holds for large n . A similar result is obtained for sums of three squares. An asymptotic formula is obtained for the related problem of representing an integer as the sum of two squares and two squares composed of small primes, as above, for any fixed θ >0. This last result is the key to bound R ( n , θ ) from below.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.350
Threshold uncertainty score0.825

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0010.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.056
GPT teacher head0.364
Teacher spread0.308 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it