Euler bricks, Euler Cuboids

Treasure Hunting Perfect Euler Bricks

Oliver knill, February 24, 2009

		Pocklington's article 1912 Spohn's article 1972 Euler Fragment E799, Postuma, 1849 Some pages of Dickson II
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.

M4V, Ogg Webm.	M4V, Ogg Webm.
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

Postscripts:

September 18 2009: Randall Rathbun writes about a connection with Heron triangles.
January 2, 2019: Updated the two movie clips from Flash to HTML5. I had also been asked about the ``close encounters" mentioned in my math table talk. Here (Mathematica.m file, labeled as .txt to be seen by the browser) is a close encounter which I still have on file.