Solutions of the congruence 𝑎^{𝑝-1}≡1 (mod 𝑝^{𝑟})
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Bibliographic record
Abstract
To supplement existing data, solutions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Superscript p minus 1 Baseline identical-to 1 left-parenthesis mod p squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>p</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <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>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">a^{p-1} \equiv 1 \pmod {p^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are tabulated for primes <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a comma p"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">a, p</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="100 greater-than a greater-than 1000"> <mml:semantics> <mml:mrow> <mml:mn>100</mml:mn> <mml:mo>></mml:mo> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>1000</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">100 > a > 1000</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="10 Superscript 4 Baseline greater-than p greater-than 10 Superscript 11"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>4</mml:mn> </mml:msup> <mml:mo>></mml:mo> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">10^4 > p > 10^{11}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . For <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a greater-than 100"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:mn>100</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a > 100</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , five new solutions <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p greater-than 2 Superscript 32"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>32</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p > 2^{32}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are presented. One of these, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals 188748146801"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mn>188748146801</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = 188748146801</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a equals 5"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>=</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">a = 5</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , also satisfies the “reverse” congruence <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p Superscript a minus 1 Baseline identical-to 1 left-parenthesis mod a squared right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <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>a</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">p^{a-1} \equiv 1 \pmod {a^2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . An effective procedure for searching for such “double solutions” is described and applied to the range <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a greater-than 10 Superscript 6"> <mml:semantics> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>></mml:mo> <mml:msup> <mml:mn>10</mml:mn> <mml:mn>6</mml:mn> </mml:msup> </mml:mrow> <mml:annotat
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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