One of the unsolved problems in mathematics. A number theory question:

Does there exist a rectangular box whose every edge and diagonal has integer length?

  • By a rectangular box, I mean a six-sided polyhedron whose sides are all rectangles.
  • The diagonals of a box include the side diagonals and the main diagonal.
  • A side diagonal joins opposite corners of a side.
  • A main diagonal joins opposite corners of a box.
To date, no such rational boxes are known to exist. It is an open question as to whether any exist at all. As a side note, it is trivially easy to come up with an integer main diagonal. Let a be an odd number. Then a^2 + ((a^2-1)/2)^2 is the square of (a^2+1)/2, which is odd. Then (((a^2+1)/2)^2-1)/2 and (((a^2+1)/2)^2+1)/2 are the other two numbers of a pythagorean triple. So the length of the diagonal would be (((a^2+1)/2)^2+1)/2 (if it were formed in this way), and the equation would look like:
a^2+((a^2-1)/2)^2+((((a^2+1)/2)^2-1)/2)^2 = ((((a^2+1)/2)^2+1)/2)^2

C. Stanley Ogilvy, Tomorrow's Math: Unsolved Problems for the Amateur. Oxford University Press. New York: 1972. Page 120.

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