Rapid multiplication modulo the sum and difference of highly composite numbers
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Bibliographic record
Abstract
We extend the work of Richard Crandall et al. to demonstrate how the Discrete Weighted Transform (DWT) can be applied to speed up multiplication modulo any number of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a plus-or-minus b"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo> ± </mml:mo> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a \pm b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="product Underscript p vertical-bar a b Endscripts p"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo> ∏ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>a</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\prod _{p|ab}{p}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is small. In particular this allows rapid computation modulo numbers of the form <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k dot 2 Superscript n plus-or-minus 1"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo> ⋅ </mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> <mml:mo> ± </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">k \cdot 2^n \pm 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In addition, we prove tight bounds on the rounding errors which naturally occur in floating-point implementations of FFT and DWT multiplications. This makes it possible for FFT multiplications to be used in situations where correctness is essential, for example in computer algebra packages.
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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) | 0.000 | 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