An old and new approach to Goormaghtigh’s equation
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A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n greater-than-or-equal-to 3"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ≥ </mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n \geq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a fixed integer, then there exists an effectively computable constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">c (n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="y"> <mml:semantics> <mml:mi>y</mml:mi> <mml:annotation encoding="application/x-tex">y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are integers satisfying <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartFraction x Superscript m Baseline minus 1 Over x minus 1 EndFraction equals StartFraction y Superscript n Baseline minus 1 Over y minus 1 EndFraction comma y greater-than x greater-than 1 comma m greater-than n comma"> <mml:semantics> <mml:mrow> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>m</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>n</mml:mi> </mml:msup> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>y</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mspace width="thickmathspace"/> <mml:mi>y</mml:mi> <mml:mo>></mml:mo> <mml:mi>x</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace"/> <mml:mi>m</mml:mi> <mml:mo>></mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \frac {x^m-1}{x-1} = \frac {y^n-1}{y-1}, \; \; y>x>1, \; m > n, \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="gcd left-parenthesis m minus 1 comma n minus 1 right-parenthesis greater-than 1"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">gcd</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>m</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\gcd (m-1,n-1)>1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="max left-brace right-brace comma comma x comma y comma m greater-than c left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo movablelimits="true" form="prefix">max</mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo>></mml:mo> <mml:mi>c</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\max \{ x, y, m \} > c (n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . In case <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n element-of StartSet 3 comma 4 comma 5 EndSet"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mn>3</mml:mn> <mml:mo>,</mml:mo> <mml:mn>4</mml:mn> <mml:mo>,</mml:mo>
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) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.001 |
| 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