Pairs of Reciprocal Quadratic Congruences Involving Primes
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Bibliographic record
Abstract
Using Pell equations and known solutions that involve Lucas sequences, we find all solutions of the reciprocal pair of quadratic congruences p2 ≡ ±1 (mod q), q2 ≡ ±1 (mod p) for odd primes p, q. In particular, we show that there is exactly one solution (p, q) = (3, 5) when the right-hand sides are -1 and 1. When the right-hand sides are both -1, there are four known solutions, all of them pairs of Fibonacci primes, and when the right-hand sides are both 1, there are no solutions. By partly different methods we completely characterize the solutions of p2 ≡ ±N (mod q), q2 ≡ ±N (mod p) for N = 2 and 4, and give partial results for N = 3 and 5. In the process we indicate how the general case can be treated.
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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 | 0.000 |
| 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 | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it