Actually you haven't. It is perfectly OK to have discussions and exhibit different opinions. It is also perfectly OK to expose factual errors in the reasoning of others. (Who would know that better than me who has been - rightfully - corrected over and over again here, most oftenly by Don Cragun.)

I think that, even restricting yourself to integer arithmetic and built-ins, you could speed up the process by using better algorithms. For instance, an application of Newtons method of caluclating the roots of differentiable functions (also called "Netwon-Raphson-method):

Suppose some real function f: [a,b] -> R, which is differentiable everywhere on the intervall [a,b] with values only in R.

It can easily be shown that, starting from an initial guess x[n] (as long as x[n] is reasonably close to x), a better guess x[n+1] can be calculated using the formula

Code:

                   f( x  )
                       n
    x     = x  - -------------
      n+1    n     f'( x  )
                        n

by solving for the tangents equation at f(x)=y and then solving for the x-intercept of this line (Simpsons approximation).

Applying this to get the zeroes of the iteration function f(x) = x**2 - a brings us to the "babylonian method" or "Heron's method" of calculating roots:

Because of the deriative f'(x) = 2x for the solution sqrt(a) we get the approximation

Code:

                    2                 
                   x  - a            
                    n         1          a  
   x     := x  - --------- = --- ( x  + ---- )
     n+1     n    2 x         2     n    x
                     n                    n

by which we calculate a series of xi's (i=0,1,2,...), which will converge against x, until a sufficient approximation is reached.

kshs floats are double-precision, as described at KSH-93 - The KornShell Command and Programming Language:
I hope this helps.

bakunin