Treasure Hunting Perfect Euler Bricks

Oliver knill, February 24, 2009

Lecture notes [PDF] of talk of February 24, 2009 Treasure map for the talk, Endpaper in HCMR, Fall 2008 [PDF of entire issue]. Navigation guide
An Euler brick is a cuboid with integer side dimensions such that the face diagonals are integers. Already in 1740, families of Euler bricks have been found. Euler himself constructed more families. If the space diagonal of an Euler brick is an integer too, an Euler brick is called a perfect Euler brick. Nobody has found one. There might be none. Nevertheless, it is an entertaining sport to go for this treasure hunt and search --- of course with the help of computers.
Scene from the Disney 1950 Movie: "Treasure Island". Here is a problem to this movie: can you prove that the Diophantine equation
   5 x4 = 3 + 7 y4 
has no integer solutions? Hint: look at the equation modulo some number n and listen to Long John Silver's parrot.
Scene from the Disney 1950 Movie: "Treasure Island". The treasure map. Pieces of Eight: Spanish Dollars