Some backstory, forgive me for not having any kind of mathematical background:
A friend of mine received an optional programming assignment to write an algorithm to find the four squares that sum to a given 5-digit number. The professor asserted that for every number there will be a solution. However, he added the constraint that all numbers must be unique amongst themselves.
I started playing with it, and for both the recursive and loop-based function I wrote, I got the following list of counterexamples when uniqueness was enforced:
12288
16384
18432
24576
28672
32768
45056
49152
65536
73728
98304
Is this an error in my programming, or does the conjecture indeed not hold when this constraint is included?