The unsolved problem of whether a perfect cuboid exists or not, or whether there is a proof pro or contra its existence... See for example http://unsolvedproblems.org/index_files/PerfectCuboid.htm
I believe to have a simple proof using logic and Reductio ad absurdum for the assumption that there is a smallest solution.
The 10th (or so) try to explain my reasoning is here:
- English cuboid-x.pdf.
- German cuboid-x.de.pdf.
- Polish cuboid-x.pl.pdf.
This little program creates perfect Pythogarean triples by applying a sequence of one out of three build rules starting with the first PPT x = 3, y = 4 and z = 5.
A pythagorean triplet (PT) consists of three natural numbers x, y and z with x^2 + y^2 = z^2. PTs with greatest common divisor 1 (PPT) are of particular interest.
Theorem 1: Every PT can be obtained in a unique way as a product of a PPT and a natural number k.
Theorem 2: In every PPT ( x | y | z ) one of the numbers x or y (legs) is even and the other one is odd. Let x always be the odd leg.
Theorem 3: For every PPT ( x | y | z ) there exists one and only one pair ( m | n ) of relatively prime natural numbers of different parity, i.e. one of the numbers is even and the other is odd, with m < n, such that:
x = n^2 - m^2, y = 2mn , z = n^2 + m^2
resp.
1/2(z - x) = m^2, 1/2(z + x) = n^2.
Theorem 4: Every PPT can in a unique way be obtained from ( 3 | 4 | 5 ) by application of a sequence of transformations A, B or C defined by:
A: (x|y|z) --> ( x-2y+2z| 2x-y+2z| 2x-2y+3z) [or (m|n) --> (n|2n-m)]
B: (x|y|z) --> ( x+2y+2z| 2x+y+2z| 2x+2y+3z) [or (m|n) --> (n|2n+m)]
C: (x|y|z) --> (-x+2y+2z|-2x+y+2z|-2x+2y+3z) [or (m|n) --> (m|2m+n)]