$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{x^2+xy+y^2}$

Question

fine $x,y$ :

$$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{x^2+xy+y^2} \ \ \ : x ,y \in \mathbb{Z}$$

My Try :

$$\sqrt{x^2-x+1}+\sqrt{y^2-y+1}=\sqrt{x^2+xy+y^2} \\ x^2-x+1+y^2-y+1+2\sqrt{(x^2-x+1)(y^2-y+1)}=x^2+xy+y^2 \\ 2\sqrt{(x^2-x+1)(y^2-y+1)}=xy +x+y-2$$

Now ?

Answer 1

You can simplify as follows

$$\begin{align} 2\sqrt{(x^2-x+1)(y^2-y+1)}&=xy +x+y-2\\ 4(x^2-x+1)(y^2-y+1)&=x^2y^2+x^2+y^2+4+2x^2y+2xy^2-4xy\\&\,\,\,\,\,+2xy-4x-4y\\ 4x^2y^2-4x^2y+4x^2-4xy^2+4xy+4y^2&=x^2y^2+2x^2y+x^2+2xy^2-2xy+y^2\\ 3x^2y^2-6x^2y+3x^2-6xy^2+6xy+3y^2&=0\\ x^2y^2-2x^2y+x^2-2xy^2+2xy+y^2&=0\\ (xy-x-y)^2&=0 \end{align}$$

From here you can get that either $x(y-1)=y$ or $y(x-1)=x$ - either way gives the same solution of $(x,y)=(2,2)$.

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As pointed out in a comment by John Bentin, $(0,0)$ is also a solution to this equation. However this is not a solution to the original equation - we have squared some terms which has created this extra solution. So it is always wise to go back and check if the solution does indeed work - $(2,2)$ does, $(0,0)$ does not.

Answer 2

Applying Minkowski's inequality, we have that $$ \sqrt{x^{2}-x+1}+\sqrt{y^{2}-y+1}=\big((x-\frac{1}{2})^{2}+\big(\frac{\sqrt{3}}{2}\big)\big)^{\frac{1}{2}}+\big((y-\frac{1}{2})^{2}+\big(\frac{\sqrt{3}}{2}\big)\big)^{\frac{1}{2}} \geq \big(\big(x-\frac{1}{2}+y-\frac{1}{2}\big)^{2}+(\sqrt{3})^{2}\big)^{\frac{1}{2}}=\sqrt{x^{2}+y^{2}+2xy-2x-2y+4} $$ Therefore, using the hypothesis, we get that $$ \sqrt{x^{2}+xy+y^{2}} \geq \sqrt{x^{2}+y^{2}+2xy-2x-2y+4} $$ which after squaring and some computation reduces to $ (x-2)(y-2) \le 0 $.

We apply Minkowsi again but this time in the form $$ \sqrt{x^{2}-x+1}+\sqrt{y^{2}-y+1}=\big(\big(x-\frac{1}{2})^{2}+\big(\frac{\sqrt{3}}{2}\big)^{2}\big)^{\frac{1}{2}}+\big(\big(\frac{1}{2}-y)^{2}+\big(\frac{\sqrt{3}}{2}\big)^{2}\big)^{\frac{1}{2}} \ge \big((x-y)^{2}+(\sqrt{3})^{2}\big)^{\frac{1}{2}} $$ Hence $$ \sqrt{x^{2}+xy+y^{2}} \geq \sqrt{x^{2}+y^{2}-2xy+3} $$ Squaring again, we get that $ xy \ge 1 $. The condition $ (x-2)(y-2) \le 0 $ together with $ xy \ge 1 $ tell us that one of $ x $ and $ y $ must necessarily be $ 1 $ or $ 2 $. WLOG, suppose $ x \in \{1,2\} $. We plug in the value of $ x $ in the original equation and computation then shows that the only possible solution is $ x=y=2 $ and this indeed satisfies the initial equation.