We shall prove it by contradiction.
Let the smallest solution of the same be (a,b,c) such that a^2 + b^2 = 3c^2\cdots(1)
Working in mod 3 we have x^2 \equiv 0 \pmod 3\cdots(2) or x^2 \equiv 1 \pmod 3\cdots(3)
in (1) RHS is divisible by 3 so LHS is also divisible by 3
For this to be true using (2) and (3) we must have
a^2 \equiv 0 \pmod 3
and b^2 \equiv 0 \pmod 3
So a and b are multiples of 3 that is a=3p and b= 3q for some p and q
Putting in (1)
3p^2 + 3q^2 = c^2 or c^2 is multiple of 3 or c is multiple of 3
So hence \frac{a}{3}, \frac{p}{3}, \frac{c}{3} is a smaller solution
Which is a contradiction
So equation does not have integer solution
This method is known as proof by infinite descent
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