As you might read on other sources, there was a computer driven proof of 2coloring the natural numbers so that no Pythagorean triple is colored equally.

Paper: https://arxiv.org/abs/1605.00723

OEIS: https://oeis.org/A272709

As many of you (including me) prefer smaller analytical proofs, I'd like to start a brainstorming, what is special about 7825. And may get a minimal subset of numbers for the proof or the main criteria why it is 7825.

7825 = 5^2 * 313

The remarkable thing on this is, that the divisors 5, 25, 313, 1565 are hypotenuse of at least one primitive Pythagorean triple:

5^2 = 4^2 + 3^2

25^2 = 24^2 + 7^2

313^2 = 312^2 + 25^2

1565^2 = 1323^2 + 836^2 = 1173^2 + 1036^2

7825 it self adds two more:

7825^2 = 1584^2 + 7663^2 = 2784^2 + 7313^2

-- v6ph1

PS: 7825 is also name of a voltage regulator chip with 25V output. - but this should not matter.