On the Infinitude of Composite NSW Numbers
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Bibliographic record
Abstract
approximately 20 years ago in connection with the order of certain simple groups. These are the numbers fn which satisfy the recurrence / » i = 6/w-/*-i 0) with initial conditions fi = l and f2 = 7. These numbers have also been studied in other contexts. For example, Bonk, Shapiro, and Simion [2] discuss them in relation to Schroder numbers and combinatorial statistics on lattice paths. Recently, Barcucci et al. [1] provided a combinatorial interpretation for the NSW numbers by defining a certain regular language 2J and studying particular properties of 2J. They close their note by asking two questions: 1. Do there exist infinitely many fn prime? 2. Do there exist infinitely many fn composite? The goal of this paper is to answer the second question affirmatively, but in a much broader context. Fix an integer k>2 and consider the sequence of values satisfying fn+l = kfn~~fn-\\> fi = l9 and f2 = k +1. Then we have the following theorem.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it