Squarefree doubly primitive divisors in dynamical sequences
Bibliographic record
Abstract
Abstract Let K be a number field or a function field of characteristic 0, let φ ∈ K ( z ) with deg( φ ) ⩾ 2, and let α ∈ ℙ 1 ( K ). Let S be a finite set of places of K containing all the archimedean ones and the primes where φ has bad reduction. After excluding all the natural counterexamples, we define a subset A ( φ , α ) of ℤ ⩾0 × ℤ >0 and show that for all but finitely many ( m , n ) ∈ A ( φ , α ) there is a prime 𝔭 ∉ S such that ord 𝔭 ( φ m + n ( α )− φ m ( α )) = 1 and α has portrait ( m , n ) under the action of φ modulo 𝔭. This latter condition implies ord 𝔭 ( φ u + v ( α )− φ u ( α )) ⩽ 0 for ( u , v ) ∈ ℤ ⩾0 × ℤ >0 satisfying u < m or v < n . Our proof assumes a conjecture of Vojta for ℙ 1 × ℙ 1 in the number field case and is unconditional in the function field case thanks to a deep theorem of Yamanoi. This paper extends earlier work of Ingram–Silverman, Faber–Granville and the authors.
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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 | 0.003 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.004 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.000 | 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; 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".