On the number of integers in a generalized multiplication table
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Bibliographic record
Abstract
Abstract. Motivated by the Erdős multiplication table problem we study the following question: Given numbers <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:msub> <m:mi>N</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> $N_1,\ldots ,N_{k+1}$ , how many distinct products of the form <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>n</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>⋯</m:mo> <m:msub> <m:mi>n</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> $n_1\cdots n_{k+1}$ with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>≤</m:mo> <m:msub> <m:mi>n</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>≤</m:mo> <m:msub> <m:mi>N</m:mi> <m:mi>i</m:mi> </m:msub> </m:mrow> </m:math> $1\le n_i\le N_i$ for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>i</m:mi> <m:mo>∈</m:mo> <m:mo>{</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>}</m:mo> </m:mrow> </m:math> $i\in \lbrace 1,\ldots ,k+1\rbrace $ are there? Call <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:msub> <m:mi>N</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $A_{k+1}(N_1,\ldots ,N_{k+1})$ the quantity in question. Ford established the order of magnitude of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mn>2</m:mn> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mi>N</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $A_2(N_1,N_2)$ and the author the one of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>A</m:mi> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mi>N</m:mi> <m:mo>,</m:mo> <m:mo>...</m:mo> <m:mo>,</m:mo> <m:mi>N</m:mi> <m:mo>)</m:mo> </m:mrow> </m:mrow> </m:math> $A_{k+1}(N,\ldots ,N)$ for all <m:math xmlns:m=
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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 | 0.004 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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