A search for Fibonacci-Wieferich and Wolstenholme primes
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
A prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called a <italic>Fibonacci-Wieferich prime</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Subscript p minus left-parenthesis StartFraction p Over 5 EndFraction right-parenthesis Baseline identical-to 0 left-parenthesis mod p squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mfrac> <mml:mi>p</mml:mi> <mml:mn>5</mml:mn> </mml:mfrac> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msub> <mml:mo> ≡ </mml:mo> <mml:mn>0</mml:mn> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">F_{p-({p\over 5})}\equiv 0\pmod {p^2}</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="upper F Subscript n"> <mml:semantics> <mml:msub> <mml:mi>F</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:annotation encoding="application/x-tex">F_n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the <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> th Fibonacci number. We report that there exist no such primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 2 times 10 Superscript 14"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> <mml:mo> × </mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>14</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p>2\times 10^{14}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . A prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p"> <mml:semantics> <mml:mi>p</mml:mi> <mml:annotation encoding="application/x-tex">p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is called a <italic>Wolstenholme prime</italic> if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartBinomialOrMatrix 2 p minus 1 Choose p minus 1 EndBinomialOrMatrix identical-to 1 left-parenthesis mod p Superscript 4 Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-OPEN"> <mml:mo maxsize="1.2em" minsize="1.2em">(</mml:mo> </mml:mrow> </mml:mstyle> <mml:mfrac linethickness="0"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:mfrac> <mml:mstyle scriptlevel="0"> <mml:mrow class="MJX-TeXAtom-CLOSE"> <mml:mo maxsize="1.2em" minsize="1.2em">)</mml:mo> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:mrow> <mml:mo> ≡ </mml:mo> <mml:mn>1</mml:mn> <mml:mspace width="0.667em"/> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em"/> <mml:msup> <mml:mi>p</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{2p-1\choose p-1}\equiv 1\pmod {p^4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . To date the only known Wolstenholme primes are 16843 and 2124679. We report that there exist no new Wolstenholme primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 10 Superscript 9"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>9</mml:mn> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p>10^9</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Wolstenholme, in 1862, proved that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartBinomialOrMatrix 2 p minus 1 Choose p minus 1 EndBinomialOrMatrix identical-to 1 left-parenthesis mod p cubed right-parenthesis"> <mml:se
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 | 0.003 | 0.001 |
| 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