The Number of Non-cyclic Sylow Subgroups of the Multiplicative Group Modulo <i>n</i>
Bibliographic record
Abstract
Abstract For each positive integer n, let $U(\mathbf {Z}/n\mathbf {Z})$ denote the group of units modulo n, which has order $\phi (n)$ (Euler’s function) and exponent $\lambda (n)$ (Carmichael’s function). The ratio $\phi (n)/\lambda (n)$ is always an integer, and a prime p divides this ratio precisely when the (unique) Sylow p-subgroup of $U(\mathbf {Z}/n\mathbf {Z})$ is noncyclic. Write W(n) for the number of such primes p. Banks, Luca, and Shparlinski showed that for certain constants $C_1, C_2>0$ , $$ \begin{align*} C_1 \frac{\log\log{n}}{(\log\log\log{n})^2} \le W(n) \le C_2 \log\log{n} \end{align*} $$ for all n from a sequence of asymptotic density 1. We sharpen their result by showing that W(n) has normal order $\log \log {n}/\log \log \log {n}$ .
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 0.000 |
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".