Bibliographic record
Abstract
Power residue symbols and their reciprocity laws have applications not only in number theory, but also in other fields like cryptography. A crucial ingredient in certain public key cryptosystems is a fast algorithm for computing power residue symbols. Such algorithms have only been devised for the Jacobi symbol as well as for cubic and quintic power residue symbols, but for no higher powers. In this paper, we provide an efficient procedure for computing 7th power residue symbols. The method employs arithmetic in the field ℚ(ζ), with ζ a primitive 7th root of unity, and its ring of integers ℤ[ζ]. We give an explicit characterization for an element in ℤ[ζ] to be primary, and provide an algorithm for finding primary associates of integers in ℤ[ζ]. Moreover, we formulate explicit forms of the complementary laws to Kummer's 7th degree reciprocity law, and use Lenstra's norm-Euclidean algorithm in the cyclotomic field.
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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 | 0.001 | 0.004 |
| 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 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.010 | 0.007 |
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".