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 distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.011 | 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; both teacher heads agree on what is shown here.
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".