Ecuatie in N-{0}(Test Sel. Franta 15.01.2013)

[mihai miculita]

Aratati ca ecuatia: $x(x+2)=y(y+1)$ nu are solutii in $x,y\in\mathbb{N}^*.$

[Pricope Tidor-Vlad]

Hint:relatia din enunt este echivalenta cu:$(2x+2)^2=(2y+1)^2+3$.

[mihai miculita]

Solutia mea:
$x(x+2)=y(y+1)\Leftrightarrow x^2+2x-y(y+1)=0\Rightarrow$$\Delta=4+4y(y+1)=4y^2+4y+4;$
asa ca: $(2y+1)^2<\Delta<(2y+2)^2\Rightarrow x\in\mathbb{R}-\mathbb{Q};\,(\forall)y>0.$
Cu alte cuvinte ecuatia data nu are nici solutii rationale pozitive!