The Challenge and aim of this project is to find four squares whose difference in each is again a square
This challenge is inspired by the question "Four squares such that the difference of any two is a square?" on Math Stack Exchange.
I published my first trials, code and ideas on Math Stack Exchange using the post "For integers x<y<z, why are these cases impossible for Mengoli's Six-Square Problem?" and got very helpful input for further investigation.
This directed me to search for 6-tuples [s,t,u,t+u,t+u−s,t−s] where each element is a square.
I implemented a Python script pythagorean_gendata2_nofile_jit.py which systematically searched for those 6-tuples and it found a lot of instances. Unfortunatelly at a certain point it became slow and did not scale anymore.
With the help of Arty, who developed an enhaced version, I could manage to find a first match. This breakthrough origins from the question on Stack Overflow Optimizing an algorithm with 3 inner loops for searching 6-tuples of squares using Python.
In order to make the final step, namely to identify these four squares, I implemented the Mathematica Script pythagorean.nb, which outputs an "almost solution": [w=40579, x=-58565, y=-65221].
From the Data Set pythagorean_stu_Arty_.txt I selected the corresponding row:
42228, 51060, 185472, 1783203984, 2607123600, 34399862784, 37006986384, 35223782400, 823919616
Hence [s=42228^2,t=51060^2,u=185472^2]. To calculate the last variable [z], I need to use the equation u+y^2=z^2 leading to z=196605.294. The last trick is to multiply all integers [w,x,y,z] with 100^2=10000, which brings us closer to a solution:
w=405790000
x=585650000
y=652210000
z=1966052940