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Friday, April 21, 2023

2023/012) prove that there is no integer solution to x^2 + y^2 = 3z^2

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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