قرار دهید:
$ \phi (x):=[ \sqrt{1} ]+[ \sqrt{2} ]+[ \sqrt{3} ]...+[ \sqrt{x^2-1} ]$
$ \Rightarrow \phi (x+1)=[ \sqrt{1} ]+[ \sqrt{2} ]+[ \sqrt{3} ]...+[ \sqrt{x^2-1} ]+[x]+[ \sqrt{x^2+1]}+...$
$+[ \sqrt{(x+1)^2-1} ] = \phi (x)+ \sum _{k=0}^{2x}[ \sqrt{x^2+k} ]=\phi (x)+ \sum _{k=0}^{2x}x($چرا؟$)$
$ \phi (x+1)= \phi (x)+x(2x+1) \Rightarrow \phi (x+1)- \phi (x)=2x^2+x, \phi (1)=0$
$\Rightarrow \phi (x)= \frac{(x-1)x(2x-1)}{3}+ \frac{x(x-1)}{2} \Rightarrow 2x(x-1)(2x-1)+3x(x-1)=6y$
$x(x-1)(4x+1)=6y \Rightarrow (x,y)=(2,3) \vee (3,13)($چرا؟$)$
$ \Box $
