infinite solutions

[Tahmid Hasan]

prove or disprove:the equation below
$x(x+1)+y(y+1)=z(z+1)$
has infinite solutions.
P.S.while solving the problem,i had a deja vu,as if i had seen it before.If it is please give me the link.

[Tahmid Hasan]

i forgot to mention that $x,y,z$ are natural numbers.

[Masum]

prove or disprove:the equation below
$x(x+1)+y(y+1)=z(z+1)$
has infinite solutions.
P.S.while solving the problem,i had a deja vu,as if i had seen it before.If it is please give me the link.

The answer is positive.
Set $x=y$ and then we get the equation :
\[z^2+z-2(x^2+x)=0\]
It will have solutions if the discriminant $8(x^2+x)+1$ is a perfect square. And the Pell equation \[t^2-2(2x+1)^2=-1\] has infinite solutions with the smallest $(t_0,x_0)=(7,2)$

[Tahmid Hasan]

i was thinking of something else.dividing the equation by $2$ shows that the sum of summation of first $x$ and $y$ positive integers is equal to the sum of first $z$ positive integers.it gets diverted into a brand new problem and it can be solved easily(easy is a relative term).