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| 第一著者: | |
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| フォーマット: | Recurso digital |
| 言語: | |
| 出版事項: |
Zenodo
2026
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| 主題: | |
| オンライン･アクセス: | https://doi.org/10.5281/zenodo.19911486 |
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- <p>Hi everyone,</p> <p>I’ve been exploring the perfect cuboid problem computationally and wanted to share some observations and get feedback.</p> <p>Quick recap: a perfect cuboid would be a box with integer edges (a, b, c) where all three face diagonals AND the space diagonal are integers. No example is known.</p> <p>In my experiments, I focused on two things:</p> <p> </p> <p>1. Modular constraints (mod 19)</p> <p>I computed:</p> <p>S = a^2 + b^2 + c^2</p> <p>For a perfect cuboid, S would have to be a perfect square.</p> <p>Looking at S mod 19, squares modulo 19 can only be:</p> <p>0, 1, 4, 5, 6, 7, 9, 11, 16, 17</p> <p>So if S mod 19 is NOT one of those values, it can’t be a perfect square → meaning that Euler brick can’t be extended to a perfect cuboid.</p> <p>I found that many Euler bricks get eliminated immediately this way.</p> <p>However, this is not a complete obstruction. Some examples still pass (for example, one case gives S ≡ 17 mod 19).</p> <p> </p> <p>2. Gap behavior</p> <p>I also looked at how close S gets to a perfect square.</p> <p>Define:</p> <p>gap = distance from S to the nearest square</p> <p>What I observed:</p> <p> </p> <p> </p> <p>Some cases have small gaps</p> <p> </p> <p> </p> <p>But there is no consistent pattern of S getting closer and closer to a square</p> <p> </p> <p> </p> <p>The behavior is irregular across different constructions</p> <p> </p> <p> </p> <p> </p> <p>Conclusion (so far)</p> <p> </p> <p> </p> <p>Modular constraints eliminate a large number of candidates</p> <p> </p> <p> </p> <p>Gap behavior doesn’t show clear convergence</p> <p> </p> <p> </p> <p>I don’t see an obvious structural path toward a perfect cuboid in the data</p> <p> </p> <p> </p> <p> </p> <p>I wrote this up more cleanly here:</p> <p>[PUT YOUR PDF LINK HERE]</p> <p> </p> <p>I’d really appreciate feedback — especially:</p> <p> </p> <p> </p> <p>Are these observations already well-known?</p> <p> </p> <p> </p> <p>Are there stronger modular frameworks people use for this problem?</p> <p> </p> <p> </p> <p>Is there a better way to approach the gap behavior?</p> <p> </p> <p> </p> <p>Thanks!</p>