Corrigendum to “Period of the power generator and small values of Carmichael’s function”
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Bibliographic record
Abstract
We are indebted to Kelly Postelmans whose question drew our attention to a slip in the proof of Theorem 8 of [1]. In particular, we asserted that for a fixed number n, the number of pairs of primes p, l with gcd(p − 1,l − 1) <Dand λ(λ(pl)) = n is at most Dτ(n), an assertion which now seems unjustified. (The notation is defined below.) In this note we give a corrected proof of Theorem 8. As in [1] we consider the power generator (1) un ≡ u e n−1 (mod m), 0 ≤ un ≤ m − 1, n =1, 2,..., with the initial value u0 = ϑ (an integer coprime to m) andexponent e (an integer at least 2). We recall that for an integer n ≥ 1theCarmichael function λ(n) is the largest order occurring amongst elements of the unit group in the residue ring modulo n. As usual, ϕ denotes Euler’s function. We let τ(n) denote the number of natural divisors of n, weletω(n) denote the number of divisors of n that are prime, and we let Ω(n) denote the number of divisors of n that are (either a prime or) a prime power. An integer n is said to be squarefull if for each prime p|n we
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 it