One could image the editors being just a little bit 'happier' with the proof if just a bit more information was provided about the number 7825.

Correct me if I'm wrong, but 7825 has to be part of at least one Pythagorean Triple, no? If you take all the integers up to 7824 and you can divide them up, but then you fail when you add 7825, then 7825 has to be part of a triple, otherwise it wouldn't be a tipping point.

So there has to be at least one set of numbers a and b such that a + b = 7825. a and b must be smaller than 7825 which is why 7825 has to be the c in the Pythagorean theorem.

So if any such numbers a and b could be given as additional information, the reason for 7825 would be clearer.

Like if you're dividing red and blue marbles over 2 containers and you can't have two of the same color in a container. You can divide 4 of them, 1 of each color in each container, but when you try to place a fifth, all containers are full.

Here, same thing: when you reach 7825, all the containers are full and you can't squeeze in an extra triple anywhere.