A prime analogue of the Erdös--Pomerance conjecture for elliptic curves
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Bibliographic record
Abstract
Let E/{\mathbb Q} be an elliptic curve of rank \ge 1 and b\in E({\mathbb Q}) a rational point of infinite order. For a prime p of good reduction, let g_b(p) be the order of the cyclic group generated by the reduction \bar b of b modulo p . We denote by \omega(g_b(p)) the number of distinct prime divisors of g_b(p) . Assuming the GRH, we show that the normal order of \omega(g_b(p)) is \log \log p . We also prove conditionally that there exists a normal distribution for the quantity \frac{\omega(g_b(p)) - \log \log p}{\sqrt{\log \log p}}. The latter result can be viewed as an elliptic analogue of a conjecture of Erdös and Pomerance about the distribution of \omega(f_a(n)) , where a is a natural number > 1 and f_a(n) the order of a modulo n .
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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 | 0.001 | 0.001 |
| 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 | 0.000 |
| 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 | 0.000 |
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 it