Bibliographic record
Abstract
Let Ln denote the largest strong Goldbach number generated by the n-th prime Pn, in this paper, we present that there are approximate bounds of Ln such that 2nlogn + 2nloglogn – 2n < Ln < 2nlogn + 2nloglogn for n ≥ 20542, based on results about prime number theorem. Let ξ(n) = Ln/2 denote the number of strong Goldbach numbers generated by Pn, equivalently, there are approximate bounds of ξ(n) such that nlogn + nloglogn – n < ξ(n) < nlogn + nloglogn for n ≥ 20542. The approximate bounds of Ln have been verified for 20542 ≤ n ≤ 400000000, equivalently, the approximate bounds of ξ(n) have also been verified for 20542 ≤ n ≤ 400000000. It is obvious that if it can be proven that there is an integer k > 0 such that bounds of 2Pn can be thought as approximate bounds of Ln for all n > k, or equivalently, there is an integer k > 0 such that bounds of Pn can be thought as approximate bounds of ξ(n) for all n > k, then Goldbach conjecture is true. We also considered another approach to the conjecture, that is, if it can be proven by introducing Li(n) that there are infinitely many Goldbach steps, then Goldbach conjecture is true.
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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 | 0.009 | 0.002 |
| 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 | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.003 |
| Insufficient payload (model declined to judge) | 0.002 | 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 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".