a^2 + b^2 = 3c^2 has no integral solutions can be proved with the algebra of odd and even:

o+o=e, e+e=e, e+o=o

oxo=o, exe=e, exo=e

Substituting (a,b,c) odd or even into original equation gives:

1) (o,o,o) → e=o, (o^2 +o^2 = 3o^2)

2) (o,e,e) → o=e

3) (e,e,0) → e=o

4) (e,e,e) → e=e

5) (o,o,e) → e=e

6) (o,e,o) → o=o

1), 2) and 3) ruled out because e=o.

4) ruled out because dividing by 2^2 (again if nec) gives one of the other cases.

5) let a=2l+1, b=2m+1, c=2n. Then,

2l^2 + 2l + 2m^2 + 2m = 6n^2 – 1 → e=o

6) let a=2l+1, b=2m+1, c=2n +1. Then,

4l^2 + 4l + 4m^2+4m = 12n^2 +12n +1 → e=o

Since that exhausts the possibilities for an integral solution, there isn’t any.