<s>[INST] There are triads of perfect squares that are consecutive terms of arithmetic progression? Prove that there exist infinitely many triples of positive integers $ x , y , z $ for which the numbers $ x(x+1) , y(y+1) , z(z+1) $ form an increasing arithmetic progression.
$ \bigg( $ It is equivalent to find all triples of $ 4x(x+1)+1=(2x+1)^{2} , 4y(y+1)+1=(2y+1)^{2} , 4z(z+1)+1=(2z+1)^{2} $ $ \bigg) $
Note : I know $ \big( 1^{2} , 5^{2} , 7^{2} \big) $ , $ \big( 7^{2} , 13^{2} , 17^{2} \big) $ , $ \big( 7^{2} , 17^{2} , 23^{2} \big) $ , $ \big( 17^{2} , 25^{2} , 31^{2} \big) $ , but how i can found all triples ?
 [/INST] We have $$x^2+x+z^2+z=2(y^2+y)$$ or  $$(x+z+1)^2+(x-z)^2=(2y+1)^2$$
We got Pythagorean triples.
Let $x+z+1=m^2-n^2$ and $x-z=2mn$, where $m>n$ and $m$ and $n$ have a different parity.
Thus, $$(x,y,z)=\left(\frac{m^2-n^2+2mn-1}{2},\frac{m^2+n^2-1}{2},\frac{m^2-n^2-2mn-1}{2}\right).$$
Also, we can always take $\frac{m^2-n^2-2mn-1}{2}>0.$
For example, for $m=6k$ and $n=2k-1$, where $k$ is a positive integer, we obtain:
$$(x,y,z)=(28k^2-4k-1,20k^2-2k,4k^2+8k-1).$$
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