Euler Bricks

Figure 1

Having posted several times about "sphenic bricks", I came across the term "euler bricks" today in the context of the number \( 25704 = 2^3 \times 3^3 \times 7 \times 17 \) which represents my diurnal age. Here is the entry in OEIS A031173 to which 25704 belongs:

Longest edge \(a \) of smallest (measured by the longest edge) primitive Euler bricks such that \( a \), \( b \), \(c\), \( \sqrt {a^2 + b^2}, \sqrt {b^2 + c^2}, \sqrt {a^2 + c^2}\) are all integers. 

240, 275, 693, 720, 792, 1155, 1584, 2340, 2640, 2992, 3120, 5984, 6325, 6336, 6688, 6732, 8160, 9120, 9405, 10725, 11220, 12075, 13860, 14560, 16800, 17472, 17748, 18560, 19305, 21476, 23760, 23760, 24684, 25704

Figure 1 shows the list of primitive numbers, up to 25704, with \(c<b<a\). In the case of 25704, the face diagonals are 31080, 17497, and 25721. Looking at Figure 1, it can be seen that there are five primitive Euler bricks with dimensions under 1000 and ten if primitive and non-primitive are included. It can be shown that if \(a\), \(b\) and \(c\) are the sides of an Euler brick then \(ka\), \(kb\) and \(kc\) with \(k\) any positive integer also form such a brick and interestingly, the sides \(ab\), \(bc\) and \(ac\) as well. The five primitive Euler bricks are depicted in Figure 2.

Figure 2: source

A perfect Euler brick is (to quote from Wikipedia):

An Euler brick whose space diagonal also has integer length. In other words, the following equation is added to the system of Diophantine equations defining an Euler brick:

\(a^{2}+b^{2}+c^{2}=g^{2}\) where \(g\) is the space diagonal. As of July 2019, no example of a perfect cuboid had been found and no one has proven that none exist.

Figure 3 illustrates this:

Figure 3: source

To quote again from Wikipedia:

An almost-perfect cuboid is defined as a cuboid that has 6 out of the 7 lengths as rational. Such cuboids can be sorted into three types, called Body, Edge, and Face cuboids. In the case of the Body cuboid, the body (space) diagonal \(g\) is irrational. For the Edge cuboid, one of the edges \(a\), \(b\), \(c\) is irrational. The Face cuboid has just one of the face diagonals \(d\), \(e\), \(f\) irrational. The Body cuboid is commonly referred to as the Euler cuboid in honor of Leonard Euler, who discussed this type of cuboid. He was also aware of Face cuboids, and provided the (104, 153, 672) example.

The smallest solutions for each type of almost-perfect cuboids, given as edges, face diagonals and the space diagonal (\(a, b, c, d, e, f, g\)):

• Body cuboid: (44, 117, 240, 125, 244, 267, √73225)
• Edge cuboid: (520, 576, √618849, 776, 943, 975, 1105)
• Face cuboid: (104, 153, 672, 185, 680, √474993, 697)