Question

Prove that the system of equations

$\{\begin{array}{cc}{x}^2+6y^2=z^2& \\ 6x2+y2=t2& \end{array}$

has no positive integer solutions.

• A. It has no positive integer solutions.
• B. It has positive integer solution .
• C. The equations do not have any solutions.
• D. The equations has negative integer solution .

Answer 1

The correct option is A It has no positive integer solutions.

we can assume that $gcd(x,y,z,t)=1$.
Adding up the equations yields$7(x^2+y^2)=z^2+t^2$.
The square residues modulo $7$ are $0,1,2,$ and $4$.
It is not difficult to see that the only pair of residues which add up to $0$ modulo $7$ is $(0,0)$, hence $z$ and $t$ are divisible by $7$. Setting $z=7z_1$ and $t=7t_1$ yields $7(x^2+y^2)=49(z_1^2+t_1^2)$ or $x^2+y^2=7(z_1^2+t_1^2)$.
It follows that $x$ and $y$ are also divisible by $7$. Contradicting the fact that $gcd(x,y,z,t)=1$.