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Bibliographic record
Abstract
CongruencesThis is an introductory chapter.The main topic is the arithmetic of congruences, sometimes called 'clock arithmetic'.It leads to the construction of the integers modulo n.These are among the simplest examples of groups, as we shall see in chapter 5.If n is a prime number, then the integers modulo n form a field.In chapter 4, we will be looking at matrices with entries in these fields.As an application of congruences we also discuss divisibility tests.In order to be able to solve linear congruences we review greatest common divisors and the Euclidean algorithm. Basic PropertiesDefinition 1.1.Fix a natural number n.The integers a and b are congruent modulo n or mod n, writtenFor example, 23 ≡ 1 (mod 11) 23 ≡ 2 (mod 7) 23 ≡ -2 (mod 25) * order 1: {1}.* order 2: Since 2 is prime, all groups of order 2 are cyclic and therefore isomorphic to Z/2Z .* order 3: Just as for 2, all groups of order 3 are isomorphic to Z/3Z .* order 4: It is easy to see that a group of order 4 is cyclic or isomorphic to V .Notice that both are abelian.* order 5: Z/5Z .* order 6: By theorem 11.10, there are two groups of order 6: Z/6Z ∼ = Z/2Z× Z/3Z and S 3 .S 3 is the smallest non-abelian group.* order 7: Z/7Z .* order 8: By theorem 11.11 there are five groups of order 8: D 4 , Q, Z/8Z , Z/4Z × Z/2Z and (Z/2Z) 3 .* order 9: From theorem 11.9 we know that every group of order 9 is either cyclic or isomorphic to ( Z/3Z ) 2 .
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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 | 0.000 | 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 | 0.000 | 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