From the arithmetic: z=2yx
From the geometric: z=y^2/x2016
Setting the RHS of each equal and setting to zero gives
y^2/x2y+x2016=0
Which is a quadratic in y.
Solving for y gives y=x(+/)x*sqrt(2016/x)
The () gives negative solutions
The (+) is easily searchable and the solutions are
(x,y,z)
(14,182,350)
(56,392,728)
(126,630,1124)
(224,896,1568)
(504,1512,2520)
(2016,4032,6048)
Since X+Y+Z=3Y these are in order.
The discriminant is asymptotic to zero and the last solution has discriminant 1 so this is the complete list.
On reflection, this could have been done analytically.
Edited on April 29, 2016, 1:16 pm

Posted by Jer
on 20160429 12:15:45 