Triangular numbers with triangular squares

[Max]

The problem is equivalent to solving the following system of Diophantine equations

1 + 8*x = y^2
1 + 8*x^2 = z^2

or, equivalently,

y^2 + z^2 = 2*(2*x+1)^2
(4*x-1)^2 + y^2 = 2*z^2

Every solution to this system deliver a pair of triangular numbers: (x,x^2)

The only known solutions are
(0,1,1)
(1,3,3)
(6,7,17)
that give respectively the following pairs of triangular numbers
(0,0)
(1,1)
(6,36)

Max

Tanya Khovanova wrote:
> Is it true that 1 and 6 are the only triangular numbers whose squares are also triangular number?
> 
> Tanya Khovanova
> 
> 
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