Bibliographic record
Abstract
We define a Gauss factorial N n ! N_n! to be the product of all positive integers up to N N that are relatively prime to n ∈ N n\in \mathbb N . In this paper we study particular aspects of the Gauss factorials ⌊ n − 1 M ⌋ n ! \lfloor \frac {n-1}{M}\rfloor _n! for M = 3 M=3 and 6, where the case of n n having exactly one prime factor of the form p ≡ 1 ( mod 6 ) p\equiv 1\pmod {6} is of particular interest. A fundamental role is played by those primes p ≡ 1 ( mod 3 ) p\equiv 1\pmod {3} with the property that the order of p − 1 3 ! \frac {p-1}{3}! modulo p p is a power of 2 or 3 times a power of 2; we call them Jacobi primes. Our main results are characterizations of those n ≡ ± 1 (
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.007 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.018 | 0.009 |
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".