Euler bricks (or Euler cuboids)
Christian Boyer, France, April 21, 2012

Even if I think that 3D perfect Euler bricks and 4D Euler bricks do not exist... here are some attempts...
Interesting results are bordered.
Open questions are in green. Feel free to send me remarks.

 In this page:

    An example of a 4D "diabolic" near-solution (693, 140, 480, 2376),
    giving 7 squares out of 8, as detailed and explained in this page:

    • 693 + 140 = 707       is a square
    • 693 + 480 = 843       is a square
    • 140 + 480 = 500       is a square
    • 693 + 2376 = 2475    is a square
    • 140 + 2376 = 116 * 421  is not a square... unfortunately...
    • 480 + 2376 = 2424    is a square
    • 693 + 140 + 480 + 2376 = 2525    its 4D diagonal is a square!
    • 693 * 140 * 480 * 2376 = 332640       the product of its edges is also a square!  

"Near-solution" means only one missing equation. "Near-miss" means that the integer is close to a squared integer in this only missing equation.


3D Euler bricks

Numerous solutions are known. For example, from his family of solutions, Euler provided (117, 44, 240):

See Wikipedia or Weisstein-MathWorld-Wolfram.

Two well-known remarks used in this page:


3D perfect Euler bricks

This system adds the 4th equation in bold to the 3D Euler brick problem: the internal diagonal has to be an integer.
It is not proved that this problem is impossible, but no solution is known: see Weisstein-MathWorld-Wolfram, this unsolved problem by Tim Roberts, and the interesting searches by Randall Rathbun and by Bill Butler.
If Rathbun's computing is correct, there is no solution with min(a, b, c) < 2.325e+10. If Butler's computing is correct, there is no solution with odd edge a < 3e+12.

For example the 3D Euler solution (117, 44, 240) is not perfect because 117 + 44 + 240 = 73225 is not a squared integer: 73225 = 5 * 29 * 101.

We can notice that the system 2 is a particular case of this other system, when k = 0:

It is possible to obtain solutions with some small k? Yes! Here is my nearest miss, a VERY nice solution (a, b, c, k) = (1887, 2943, 5247, 2). With a so small k, this is very close to a 3D perfect Euler brick!

I have found only three solutions with -10000 < k < 10000, the example above and these two others:

Solutions with small k are very rare. Is it possible to obtain other primitive solutions of [system 3] with abs(k) < 10000?

How close to an integer can the internal diagonal be? Any 3D Euler brick [system 2] can be written that way, s being the integer minimizing abs(k), with k positive or negative:

Because we would like k = 0, is it at least possible to obtain some small k? I think that the nearest miss, the best known brick having the smallest abs(k), is this one, with k = -120:

I have found only six primitive 3D Euler bricks with -1000 < k < 1000, the example above and these five other near misses:

Solutions with small k are very rare. Is it possible to obtain other primitive solutions of [system 4] with abs(k) < 1000?

The bigger the 3D bricks, the bigger their k. For example, with any s > 1e+8, it seems extremely difficult (or impossible?) to obtain abs(k) < 1e+5, the best solution being probably:

Is it possible to obtain primitive solutions of [system 4] with s > 1e+8 and abs(k) < 1e+5?

And we want k = 0... The bigger the 3D bricks, the smaller will be the probability to directly obtain k = 0, the perfect Euler brick seems hopeless...


4D Euler bricks

Today, it is unknown if a 4D Euler brick (a, b, c, d) can exist: see this unsolved problem by Tim Roberts. If they exist:

General idea on how to obtain 4D near-solutions

A 4D brick can be seen as a combination of three 3D Euler bricks having the same odd edge a:

An idea which could solve this problem is to combine two primitive 3D Euler bricks having the same ratio of two edges, then scale them in order to obtain the same edges: this will directly produce a solution of at least 5 out of the 6 equations of the system 5! If the 6th equation with the two remaining edges is not true, we will call it a "4D near-solution". If the equation is true, then we have the Grail, a 4D solution.

For example the 3D Euler solution (117, 44, 240) and this other 3D solution (429, 880, 2340) have one same ratio 117 / 240 = 429 / 880.
We will scale theses bricks by factors 11 and 3, because 11*117 = 3*429 = lcm(117, 429) =
1287.
Now 11*(117, 44, 240)=(
1287, 484, 2640) and 3*(429, 880, 2340)=(1287, 2640, 7020) have two edges in common in red and in blue, and we obtain the 4D near-solution (1287, 484, 2640, 7020).
Checking the only missing equation of the two other edges in italics, unfortunately this do not give a squared integer:

Three 4D near-solutions from any 3D brick, using the derived brick

The properties of the previous example can be explained because we have combined a brick and its derived brick. The derived brick of (117, 44, 240) is (429, 880, 2340), because (117*44, 117*240, 44*240) = (5148, 28080, 10560) = 12*(429, 2340, 880). With this brick and its derived brick, we can in fact generate three 4D near-solutions! The two others are (429, 2340, 4800), and (4563, 3520, 9360). Why three?

Each primitive 3D brick (a, b, c), combined with its derived brick (ab, ac, bc), produces three 4D near-solutions having at least 5 true equations out of 6:

  • a*(a, b, c)=(a, ab, ac)  &  (ab, ac, bc)  ->  (ab, ac, bc, a) which could become a 4D Euler brick if and only if a4 + bc is a squared integer (the 6th missing equation)
  • b*(a, b, c)=(ab, b, bc)  &  (ab, ac, bc)  ->  (ab, ac, bc, b) which could become a 4D Euler brick if and only if b4 + ac is a squared integer (the 6th missing equation)
  • c*(a, b, c)=(ac, bc, c)  &  (ab, ac, bc)  ->  (ab, ac, bc, c) which could become a 4D Euler brick if and only if c4 + ab is a squared integer (the 6th missing equation)

Maybe one of the known parametric 3D bricks (a, b, c) will allow a4+bc or b4+ac or c4+ab to be a square, and then immediately produce a 4D brick?
For example the parametric solution (x(x-3y), y(3x-y), 4xyz), with x=p-q, y=2pq, z=p+q produces Euler's solution (117, 44, 240) when p=2, q=1. Can [x(x-3y)]4 + [4xyz(3x-y)] be a square?

Proof. If (a, b, c) is a 3D Euler brick, then (ab, ac, bc, a) is a 4D near-solution because:

Similar proofs with (ab, ac, bc, b) and (ab, ac, bc, c) which are the two other near-Euler bricks.

Three more 4D near-solutions from any 3D brick, repeating an edge

Each primitive 3D brick (a, b, c) produces three 4D near-solutions, in simply repeating one of its edges:

  • (a, b, c, a)
  • (a, b, c, b)
  • (a, b, c, c)

Unfortunately none of these supplemental solutions can be a 4D brick, because twice a square cannot be a square.

However here a nice solution (a, b, c, c), very close to a 4D Euler brick. This is the nearest possible miss, a delta of only 1:

Other 4D near-solutions

Using derived bricks or repeating edges are not the only ways to obtain near-solutions. We can also obtain 4D near-solutions combining bricks and NOT-derived bricks, for example :

Another 4D near-solution, with something strange, two odd edges instead of only one...:

Computing

Using the 30414 primitive 3D solutions with smallest edge < 2.325e+10 (*) kindly provided by Randall Rathbun, and their derived solutions (even if their smallest edges become > 2.325e+10), there was a (small) hope to obtain a 4D solution. Combining them, excluding solutions repeating edges which cannot produce the 6th equation, I have constructed 93550 couples having a same ratio. Here is a zipped text file (6Mb) of 93550 primitive 4D near-solutions. By construction, all of of these solutions have 5 equations true, but unfortunately, none of them produce the 6th equation true...

Among these solutions, here is the nearest miss of a squared integer, with a delta of only 96. This solution do not come from a brick and its derived brick:

If somebody has a longer list of primitive 3D solutions, maybe this method will produce a 4D solution?

(*) Meaning that other edges are bigger, the biggest edge of this list being ~2.2e+15


4D perfect Euler bricks

This system adds the 7th equation in bold to the 4D Euler brick problem: the 4D internal diagonal has to be an integer. Because we don't know a solution of the 6 first equations, obviously we do not know a solution to these 7 equations. But if we analyze the first numerical example (1287, 484, 2640, 7020) of the previous part, the 7th equation is true!

Surprising, but in fact easy to explain. Because this solution and a lot of others come from a combination of bricks and their derived bricks, we have seen that their form is (ab, ac, bc, a) or similar. The sum of squares of their edges is ab + ac + bc + aa  = (aa + ab) + (ac + bc) = a(a + b) + c(a + b) = (a + c)(a + b) = (pq), i.e. a square!

Among the 93550 4D near-solutions computed in the previous part, there are 71553 primitive 4D perfect near-solutions (5Mb zipped text file). Here are the smallest perfect near-solutions, with 4D diagonals v < 10000:

The nearest miss of a squared integer that I have found is with (693, 140, 480, 2376), delta of 576:


4D "diabolic" Euler bricks

This system adds a 8th equation to the 4D perfect Euler brick problem: the product of the 4 edges has to be a square. Excellent news: solutions with 7 true equations out of 8 are very easy to find. All of our 7 examples with v < 10000 listed above have this last equation true! For example with (1287, 484, 2640, 7020):

Looks surprising, but again easy to explain. Because these solutions come from combinations of bricks and their derived bricks, their form is (ab, ac, bc, a) or similar. The product of their edges is ab * ac * bc * a = (abc), i.e. a square!

Here are listed the full nice characteristics of (693, 140, 480, 2376), one of these 4D "diabolic" near-solutions:

In fact, it's very difficult to find 4D perfect near-solutions having a + b + c + d = v where abcd is not a square: there are only two examples among the list of 71553!

It's also very difficult to find 4D near-solutions having abcd = w where a + b + c + d is not a square. Because I am unable to find bigger examples, here are perhaps the two only possible examples. They are derived, because with (a, b, c, d) = (1105, 9360, 29700, 35904), we can construct (abc, abc, acd, bcd) = (1105*9360*29700, 1105*9360*35904, 1105*29700*35904, 9360*29700*35904) = 105019200*(95040, 11220, 3536, 2925).

Just for fun, here the biggest 4D "diabolic" near-solution of my long list: (15607515986148064713435, 68420270979732808900, 821525407157461544149920, 187400177485923053458405368).


4D fully perfect Euler bricks

The 4D "perfect" Euler bricks check only its 2D and 4D diagonals. If we also want the 3D diagonals of 4D bricks, we will call them "fully perfect" bricks and we will have to solve:

Hmmm.... Because we do not know any 3D perfect brick, we do not know any 4D near-solution having at least p, q, r and v.


5D (or 6D or 7D...) Euler bricks

Again hmmm... However, with similar ideas combining 3D Euler bricks, we may find solutions with some of the needed equations which will be true.

For example, with any 3D Euler brick (a, b, c), we can construct a 6D (ab, ac, bc, a, b, c) which will have at least 9 equations true. With Euler's (117, 44, 240), we obtain (5148, 28080, 10560, 13689, 1936, 57600):

But unfortunately the 6 other sums are not squares: