Bibliographic record
Abstract
The similarity between prime numbers and irreducible polynomials has been a dominant theme in the development of number theory and algebraic geometry. There are certain conjectures indicating that the connection goes well beyond analogy. For example, there is a famous conjecture of Buniakowski formulated in 1854 (see Lang [3, p. 323]), independently reformulated by Schinzel, to the effect that any irreducible polynomial f (x) in Z[x] such that the set of values f (Z +) has no common divisor larger than 1 represents prime numbers infinitely often. In this instance, the theme is to produce prime numbers from irreducible polynomials. This conjecture is still one of the major unsolved problems in number theory when the degree of f is greater than one. When f is linear, the conjecture is true, of course, and follows from Dirichlet’s theorem on primes in arithmetic progressions. It is not difficult to see that the converse of the Buniakowski conjecture is true; namely, if a polynomial represents prime numbers infinitely often, then it is an irreducible polynomial. To see this, let us try to factor f (x) = g(x)h(x) with g(x) and h(x) in Z[x] of positive degree. The fact that f (x) takes prime values infinitely often
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.528 | 0.136 |
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".