Detecting perfect powers by factoring into coprimes
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
This paper presents an algorithm that, given an integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than 1"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , finds the largest integer <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> th power. A previous algorithm by the first author took time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b Superscript 1 plus o left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:msup> <mml:mi>b</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mo>+</mml:mo> <mml:mi>o</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding="application/x-tex">b^{1+o(1)}</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="b equals log base 10 n"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo>=</mml:mo> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>n</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">b=\lg n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; more precisely, time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b exp left-parenthesis upper O left-parenthesis StartRoot log base 10 b log base 10 log base 10 b EndRoot right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mi>exp</mml:mi> <mml:mo> </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msqrt> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>b</mml:mi> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>b</mml:mi> </mml:msqrt> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">b \exp (O(\sqrt {\lg b\lg \lg b}))</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ; conjecturally, time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b left-parenthesis log base 10 b right-parenthesis Superscript upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>b</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">b (\lg b)^{O(1)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . The new algorithm takes time <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b left-parenthesis log base 10 b right-parenthesis Superscript upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>b</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">b(\lg b)^{O(1)}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . It relies on relatively complicated subroutines—specifically, on the first author’s fast algorithm to factor integers into coprimes—but it allows a proof of the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b left-parenthesis log base 10 b right-parenthesis Superscript upper O left-parenthesis 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>b</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>lg</mml:mi> <mml:mo> </mml:mo> <mml:mi>b</mml:mi> <mml:msup> <mml:mo stretc
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 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