1. Finding the perfect Euler brick
Megan Grywalski

2. Euler brick
An Euler brick is a cuboid whose side lengths and face diagonals are integers.
For example, a cuboid with dimensions (a, b, c) = (44, 117, 240) is an Euler brick. It is actually the smallest Euler brick, which was found by Paul Halcke in This Euler brick has face diagonals (d, e, f) = (125, 244, 267).
An Euler brick satisfies the following equations:
a2 + b2 = d2
a2 + c2 = e2
b2 + c2 = f2
d = dab e = dac f = dbc
Must be
integers

3. A perfect Euler brick
A perfect cuboid, or perfect Euler brick, is an Euler brick whose space diagonal is also an integer.

4. It is unknown if a perfect Euler brick exists, nor has anyone
A perfect Euler brick must satisfy the equations for sides (a, b, c), face diagonals (d, e, f), and space diagonal g with a, b, c, d, e, f, and g ∈ ℤ:
a2 + b2 = d2
a2 + c2 = e2
b2 + c2 = f2
a2 + b2 + c2 = g2
d = dab e = dac f = dbc g = dabc
It is unknown if a perfect Euler
brick exists, nor has anyone
proved that one does not exist.

5. In 1740, Nicholas Saunderson, a blind mathematician, came up with a parameterization that produced Euler bricks.
If (u, v, w) is a Pythagorean triple u2 + v2 = w2 then
(a, b, c) = (|u(4v2 - w2)|, |v(4u2 - w2)|, |4uvw|)
if u = 2st, v = s2 - t2, w = s2 +t2
a =6ts5 − 20t3s3 + 6t5s, b = −s6 + 15t2s4 − 15t4s2 + t6, c = 8s5t − 8st5.
The surface r(s, t) = <6ts5 − 20t3s3 + 6t5s,−s6 + 15t2s4 − 15t4s2 + t6, 8s5t − 8st5> is an Euler brick.
a2 +b2 +c2 = f(t, s)(s2 +t2)2
f(t, s) = s8 + 68s6t s4t4 + 68s2t6 + t8,
if you found s, t for which f(t, s) is a square then this would be a perfect Euler brick.

6. Using these parameters with the help of a computer it was found that there exists an (a, b, c) with a having digits, b with digits, and c with digits so that the space diagonal √(a2 + b2 + c2) is close to an integer.

7. Exhaustive computer searches show that, if a perfect cuboid exists, one of its sides must be greater than 1012.
Solutions have been found where two of the three face diagonals and the space diagonal are integers, such as:
(a, b, c) = (672, 153, 104)
Some solutions have been found where all four diagonals but only two of the three sides are integers, such as:
(a, b, c) = (18720, √ , 7800) and
(a, b, c) = (520, 576, √618849)