Problem 2

Prove that the equation

6(6a^2+3b^2+c^2)=5n^2

has no integer solutions except

a=b=c=n=0

Step 3 of 5: Use modulo 8 to make b and c0 even

Previous step Next step

b^2+3c_0^2\equiv2(n_0^2-a^2)\pmod8\Rightarrow b,c_0\text{ are even}

Detailed analysis

Modulo $2$ , the reduced equation says $b$ and $c_0$ have the same parity. Modulo $8$ , it gives $b^2+3c_0^2\equiv2(n_0^2-a^2)$ . The right side is $0$ modulo $8$ because the same congruence first forces $a,n_0$ to have the same parity. If $b,c_0$ were odd, the left side would be $1+3\equiv4\pmod8$ , impossible. Thus $b=2b_0$ and $c_0=2c_1$ .