Bibliographic record
Abstract
In 2003, Zhao discovered a curious congruence involving harmonic series and Bernoulli numbers: for any odd prime p [Formula: see text] where [Formula: see text] is the nth Bernoulli number. This congruence was generalized by Wang and Cai in 2014, and Cai, Shen and Jia in 2017 by replacing the odd prime p in the summation and modulus with an odd prime power, and a product of two odd prime powers, respectively. In particular, Cai, Shen and Jia proposed a conjectural congruence: for any positive integer n with an odd prime factor p such that [Formula: see text] where [Formula: see text] [Formula: see text] In this paper, we establish the following generalization of their conjecture: for any positive integer n with an odd prime factor [Formula: see text] such that [Formula: see text] where [Formula: see text] [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are positive integers coprime to [Formula: see text], and [Formula: see text] is a positive common multiple of [Formula: see text], [Formula: see text] and [Formula: see text]. Also, [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].
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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 | 0.004 | 0.018 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.005 | 0.011 |
| Scholarly communication | 0.004 | 0.011 |
| Open science | 0.001 | 0.006 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.009 | 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".