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Week 20 · ClassicalOpen

Perfect Cuboid — Can All 7 Lengths Be Integers?

Simple rule · open problem· Posed 1894

Problem

Does there exist a rectangular cuboid whose edges $a, b, c$ , three face diagonals $\sqrt{a^2+b^2}, \sqrt{a^2+c^2}, \sqrt{b^2+c^2}$ , and space diagonal $\sqrt{a^2+b^2+c^2}$ are all simultaneously integers?

Why it matters

Paul Halcke recorded the smallest Euler brick $(44,117,240)$ in 1719: six integer outer lengths but a non-integer space diagonal. Artemas Martin published the perfect-cuboid question in 1894. No box satisfying all four Diophantine constraints is known.

Progress so far

Matson's computer search implies that any solution must have an odd edge above $2.5 \times 10^{13}$ and a smallest edge above $5 \times 10^{11}$ . Huge negative searches and many necessary conditions are known, but neither a solution nor a nonexistence proof.

💡 Explore together, one line at a time(0 contributions)

Contributions are not ranked by popularity. Curator feedback names what is clear or reproducible, and peer signals mean someone understood or actually reproduced it.

What did you notice?

You do not need a complete proof. A small observation can open the next path.

Markdown + KaTeX supported (`$x^2$` inline, `$$\sum_{k=1}^n k$$` display)

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