Some blocks with integer space diagonals.
12 + 22 + 22 = 32
22 + 32 + 62 = 72
42 + 42 + 72 = 92
12 + 42 + 82 = 92
62 + 62 + 72 = 112
22 + 62 + 92 = 112
32 + 42 + 122 = 132
The last one is neat because it can be seen as constructed from the 3-4-5 and 5-12-13 Pythagorean triples.
See "Euler Brick" for integer face diagonals, for example 44-117-240.
The perfect cuboid with integer space diagonals is a tantalizing problem, as one would naively expect that a general theory about the solubility of Diophantine equations of only second powers (no linear terms, no cubic terms like elliptic curves, no unknowns in the exponent like Fermat's Last Theorem) would have been developed already, or at least a meta theory about why such a theory would be difficult. Arbitrary Diophantine equations are impossible by Hilbert's 10th problem, but are mere squares equally difficult?
The Pell equation suggests how large solutions to quadratic Diophantine equations can be. A perfect cuboid may well exist.
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