MO - Puzzles (I)

❚ I stumbled upon a really nice MO puzzle. No, that's is not a puzzle for my dog, who is also named MO, but a puzzle taken from the well known Mathematical Olympiad, which takes places once every year.

Puzzle 1: Find all triples of integers $(a,b,c) \in \mathbb{N}^3_{> 0}$ such that $$\text{I)}\;ab-c = 2^{e_1}, \;\; \text{II)}\;ac-b = 2^{e_2}, \;\; \text{III)}\; bc-a = 2^{e_3}$$ with $(e_1,e_2,e_3) \in \mathbb{N}^3$

If you have some free hours, try it for yourself. I don't think that the proof below the fold is very nice or beautiful, and also for my taste distinguished too many cases, but it is overall not that hard to follow.

Proof