Classification of integers based on residue classes via modern deep learning algorithms
Bibliographic record
Abstract
Judging whether an integer can be divided by prime numbers such as 2 or 3 may appear trivial to human beings, but it can be less straightforward for computers. Here, we tested multiple deep learning architectures and feature engineering approaches to classifying integers based on their residues when divided by small prime numbers. We found that the ability of classification critically depends on the feature space. We also evaluated automated machine learning (AutoML) platforms from Amazon, Google, and Microsoft and found that, without appropriately engineered features, they failed on this task. Furthermore, we introduced a method that utilizes linear regression on Fourier series basis vectors and demonstrated its effectiveness. Finally, we evaluated large language models (LLMs) such as GPT-4, GPT-J, LLaMA, and Falcon, and we demonstrated their failures. In conclusion, feature engineering remains an important task to improve performance and increase interpretability of machine learning models, even in the era of AutoML and LLMs.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.004 | 0.002 |
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".