Pseudopowers and primality proving
Bibliographic record
Abstract
It has been known since the 1930s that so-called pseudosquares yield a very powerful machinery for the primality testing of large integers N. In fact, assuming reasonable heuristics (which have been confirmed for numbers to 2^80) this gives a deterministic primality test in time O((lg N)^(3+o(1))), which many believe to be best possible. In the 1980s D.H. Lehmer posed a question tantamount to whether this could be extended to pseudo r-th powers. Very recently, this was accomplished for r=3. In fact, the results obtained indicate that r=3 might lead to an even more powerful algorithm than r=2. This naturally leads to the challenge if and how anything can be achieved for r>3. The extension from r = 2 to r = 3 relied on properties of the arithmetic of the Eisenstein ring of integers Z[\zeta_3], including the Law of Cubic Reciprocity. In this paper we present a generalization of our result for any odd prime r. The generalization is obtained by studying the properties of Gaussian and Jacobi sums in cyclotomic ring of integers, which are tools from which the r-th power Eisenstein Reciprocity Law is derived, rather than from the law itself. While r=3 seems to lead to a more efficient algorithm than r=2, we show that extending to any r>3 does not appear to lead to any further improvements.
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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 | 0.004 | 0.006 |
| 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 | 0.000 |
| 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; a candidate call from one teacher head, 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".