Euler bricks (or Euler cuboids)
Christian Boyer, France, AprilMay 2012
(May
2020, short update with Matson's computing results published in 2015)
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" nearsolution (693, 140, 480, 2376), 
"Nearsolution" means only one missing equation. "Nearmiss" means that the integer is close to a squared integer in this only missing equation.
Numerous solutions are known. For example, from his family of solutions, Euler provided (117, 44, 240):
See Wikipedia or WeissteinMathWorldWolfram.
Two wellknown remarks used in this page:
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 WeissteinMathWorldWolfram,
this
unsolved
problem by Tim Roberts, and the interesting searches by Randall
Rathbun, by Bill
Butler and by Rob Matson.
For example the 3D Euler solution (117, 44, 240) is not perfect
because 117² + 44² + 240² = 73225 is not a squared integer: 73225
= 5² * 29 * 101.
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.
If Matson's computing is correct, there is no
solution with odd edge a < 2.5e+13.
If my computing is correct:

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...
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 nearsolutions
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 nearsolution". 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 nearsolution
(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 nearsolutions from any 3D brick, using the derived brick
The properties of the previous example (1287, 484, 2640, 7020) 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 nearsolutions! The two others are (429, 880, 2340, 4800), and (4563, 1716, 3520, 9360). Why three?
Each primitive 3D brick (a, b, c), combined with its derived brick (ab, ac, bc), produces three 4D nearsolutions having at least 5 true equations out of 6:
Maybe one of the known parametric
3D bricks (a, b, c) will allow a^{4}+b²c² or b^{4}+a²c² or c^{4}+a²b² to be a square, and then
immediately produce a 4D brick? 
Proof. If (a, b, c) is a 3D Euler brick, then (ab, ac, bc, a²) is a 4D nearsolution because:
Similar proofs with (ab, ac, bc, b²) and (ab, ac, bc, c²) which are the two other nearEuler bricks.
Three more 4D nearsolutions from any 3D brick, repeating an edge
Each primitive 3D brick (a, b, c) produces three 4D nearsolutions, in simply repeating one of its edges:
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:
(4901, 4368, 13860, 13860)
13860² + 13860² = 2*(13860)² = 19601²
 1 (= 19600.999974...)²
Other 4D nearsolutions
Using derived bricks or repeating edges are not the only ways to obtain nearsolutions. We can also obtain 4D nearsolutions combining bricks and NOTderived bricks, for example :
17*(117, 44, 240) & 4*(187,
1020, 1584) > (1989, 748, 4080, 6336)
1989² + 748² + 4080² + 6336²
= 61306921 = 13 * 761 * 6197
Another 4D nearsolution, with something strange, two odd edges instead of only one...:
3*(117, 44, 240) & (85, 132, 720) >
(85, 351, 132, 720)
85² + 351² = 130426 = 2 * 65213
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 nearsolutions. 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:
21*(101565, 240900, 1041392) & 61*(34965,
62900, 358512) > (2132865, 3836900, 5058900, 21869232)
3836900² + 5058900² = 40314270820000 =
2^5 * 5^4 * 16657 * 121013 = 6349352² + 96
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
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, a²) or similar. The sum of squares of their edges is a²b² + a²c² + b²c² + a²a² = (a²a² + a²b²) + (a²c² + b²c²) = a²(a² + b²) + c²(a² + b²) = (a² + c²)(a² + b²) = (pq)², i.e. a square!
Among the 93550 4D nearsolutions computed in the previous part, there are 71553 primitive 4D perfect nearsolutions (5Mb zipped text file). Here are the smallest perfect nearsolutions, 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:
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, a²) or similar. The product of their edges is ab * ac * bc * a² = (a²bc)², i.e. a square!
Here are listed the full nice characteristics of (693, 140, 480, 2376), one of these 4D "diabolic" nearsolutions:

In fact, it's very difficult to find 4D perfect nearsolutions having a² + b² + c² + d² = v² where abcd is not a square: there are only two examples among the list of 71553!
In the list of 71553 primitive 4D perfect nearsolutions, given in the previous part, 71551 of them are "diabolic" nearsolution, with 7 true equations out of 8. 
It's also very difficult to find 4D nearsolutions having abcd = w² where a² + b² + c² + d² 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" nearsolution of my long list: (15607515986148064713435, 68420270979732808900, 821525407157461544149920, 187400177485923053458405368).
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 nearsolution having at least p², q², r² and v².
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, a², b², c²) 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: