Bibliographic record
Abstract
Abstract In his proof of the irrationality of $\zeta (3)$ and $\zeta (2)$ , Apéry defined two integer sequences through $3$ -term recurrences, which are known as the famous Apéry numbers. Zagier, Almkvist–Zudilin, and Cooper successively introduced the other $13$ sporadic sequences through variants of Apéry’s $3$ -term recurrences. All of the $15$ sporadic sequences are called Apéry-like sequences. Motivated by Gessel’s congruences mod $24$ for the Apéry numbers, we investigate congruences of the form $u_n\equiv \alpha ^n \ \pmod {N_{\alpha }}~(\alpha \in \mathbb {Z},N_{\alpha }\in \mathbb {N}^{+})$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$ . Let $N_{\alpha }$ be the largest positive integer such that $u_n\equiv \alpha ^n\ \pmod {N_{\alpha }}$ for all non-negative integers n. We determine the values of $\max \{N_{\alpha }|\alpha \in \mathbb {Z}\}$ for all of the $15$ Apéry-like sequences $\{u_n\}_{n\ge 0}$ . The binomial transforms of Apéry-like sequences provide us a unified approach to this type of congruences for Apéry-like sequences.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
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".