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Hi, here is my code

%time
var('n,R,T')
eq1 = ((n-1)^2/(n+1)^2)==(-1-2*R+R^2-T^2+sqrt(4*T^2+(1-2*R+R^2-T^2)^2))/(2*R-4)
sol=solve(eq1,n,solution_dict=true)
test1 = (eq1.substitute(n=sol[0][n]))
test2 = test1.lhs()-test1.rhs()
test3 = expand(test2).full_simplify()
test3

and the glorious result

0
CPU time: 4575.08 s,  Wall time: 4601.30

I would like to thank DSM for pointing out that my previous attempt would not result into an integer and John Palmieri to extend the usable memory for the virtual machine.

Hi, here is my code

%time
var('n,R,T')
eq1 = ((n-1)^2/(n+1)^2)==(-1-2*R+R^2-T^2+sqrt(4*T^2+(1-2*R+R^2-T^2)^2))/(2*R-4)
sol=solve(eq1,n,solution_dict=true)
test1 = (eq1.substitute(n=sol[0][n]))
test2 = test1.lhs()-test1.rhs()
test3 = expand(test2).full_simplify()
test3

and the glorious result

0
CPU time: 4575.08 s,  Wall time: 4601.30

I would like to thank DSM for pointing out that my previous attempt would not result into an integer and John Palmieri to extend the usable memory for the virtual machine.