An Optimal Bound for Sum of Square Roots of Special Type of Integers
Bibliographic record
Abstract
Abstract The sum of square roots of integers problem is to find the minimum nonzero difference between two sums of square roots of integers. Let r(n,k) denote the minimum nonzero positive value: | ∑ k √ i=1 ai − ∑ k √ i=1 bi|, where ai and bi are positive integers not larger than integer n. We prove by an explicit construction that r(n,k) = O(n−2k+ 3 2) for fixed k and any n. Our result implies that in order to compare two sums of k square roots of integers with at most d digits per integer, one might need precision of as many as (2k − 3)d digits. We also prove that this bound is optimal 2 for a wide range of integers, i.e., r(n,k) = Θ(n−2k+ 3 2) for fixed k and for those integers in the form of n = � �2 � �2 2k−1
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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 | 0.004 | 0.030 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.004 | 0.010 |
| Open science | 0.003 | 0.003 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 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".