Bibliographic record
Abstract
Let n/d ∈ Q, m be a positive integer and let u = n/d mod m. Thus $u$ is the image of a rational number modulo m. The rational reconstruction problem is; given u and m find n/d. A solution was first given by Wang in 1981. Wang's algorithm outputs n/d when m > 2 M2 where M = max(|n|,d). Because of the wide application of this algorithm in computer algebra, several authors have investigated its practical efficiency and asymptotic time complexity.In this paper we present a new solution which is almost optimal in the following sense; with controllable high probability, our algorithm will output n/d when m is a modest number of bits longer than 2 |n| d. This means that in a modular algorithm where m is a product of primes, the modular algorithm will need one or two primes more than the minimum necessary to reconstruct n/d; thus if |n| ⇐ d or d ⇐ |n| the new algorithm saves up to half the number of primes. Further, our algorithm will fail with high probability when m < 2 |n| d.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.002 | 0.013 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".