Rational Polynomials That Take Integer Values at the Fibonacci Numbers
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Bibliographic record
Abstract
AbstractAn integer-valued polynomial on a subset S of ℤ is a polynomial f (x) ∊ ℚ [x] with the property f (S) ⊆ ℤ. This article describes the ring of such polynomials in the special case that S is the Fibonacci numbers. An algorithm is described for finding a regular basis, i.e., an ordered sequence of polynomials, the nth one of degree n, with which any such polynomial can be expressed as a unique integer linear combination. Additional informationNotes on contributorsKeith JohnsonKEITH JOHNSON received his Ph.D. in mathematics from Brandeis University and now teaches at Dal-housie University where he is a professor of mathematics. His interest in rings of integer-valued polynomials was originally sparked by their occurrence in algebraic topology.Kira ScheibelhutKIRA SCHEIBELHUT received her B.Sc. in mathematics from Dalhousie University in 2011. After a year spent traveling, she returned to Dalhousie and completed her M.Sc. in mathematics in 2013. She is currently working as an implementation analyst with the consulting firm Morneau Shepell while she contemplates yet another return to school.
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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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
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