Don,
I'm quite sleep-deprived and at the moment, so whatever I'm about to say may not make too much sense... We'll see...
First, the latex. I wish you'd invest some time in it—it's really not difficult once you get the basics and it would help everybody out. If i've followed your parentheses correctly, this might be right:
![\frac{\left[\left(sin(x^{(1/2)})\right)-1\right]^{-1}}{\pi^2+\left[\ln\left(\ln\left(2\left[\left[2(x^{-1}+1)\right]^{-1}+1\right]\right)\right)\right]^{-1}}-1=0](http://hypography.com/forums/latex/img/38faa33b0c09c540a50b72059a2ee470-1.gif "\frac{\left[\left(sin(x^{(1/2)})\right)-1\right]^{-1}}{\pi^2+\left[\ln\left(\ln\left(2\left[\left[2(x^{-1}+1)\right]^{-1}+1\right]\right)\right)\right]^{-1}}-1=0")
I put your value of x,
2.566,543,832,171,388,844,467,529
in your equation,
((sin(x^(1/2)))^(-1)-1)^(-1) / ((pi)^2 + (ln(ln(2*((2* (x^(-1) +1))^(-1) +1))))^(-1))-1
using PERL at 100 digit accuracy and got:
-0.000000000040019755626388018650144802402308556
Solving recursively gives these results:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 62 71 100
Which has diverged by 62.
The incriminating x is shown here:
Code:
int = 2, x = 2.566543832171388844467529000000000000000000000000000000000000000000000
int = 3, x = 3.530176989721365539402422018201306850039197136533050911036155166398856
int = 5, x = 5.658487746849688216649061411000589507461664004153497197119476215483240
int = 7, x = 7.593155717658844724384335263645162617947202893363521789006038087166225
int = 11, x = 11.80128555049362404460174832837998655716729063627250228160779891806179
int = 13, x = 13.72794004986556753523763397429744246725502028275445142599165199144844
int = 17, x = 17.85861349763730877250927709413576433988653038229942139918833510613291
int = 19, x = 19.79936263147479092664292299016437987747272027540063908052642739544050
int = 23, x = 23.76893696537426002453788600930707468800883412711664132116378903885593
int = 29, x = 29.91142451944074443786296806872299704308061095064639403832115527561836
int = 31, x = 31.81832327464107973441092377400036596929785102069880661037360711858067
int = 37, x = 37.88233936471712759962716274091303887571471394483018993191178809579994
int = 41, x = 41.93397855694896393020755651340059733680483703198842232377858911936599
int = 43, x = 43.89822410263364625518520937029838556476463991279448629612635446175649
int = 47, x = 47.87224020589233116148280327819598753456306848077071626646202280455230
int = 53, x = 53.88423932134315471134298755419452646380031791291086201039183094297123
int = 59, x = 59.93852424419815301864134307323740915524877993867733325427721822943831
int = 62, x = 62.86465199458734434295017417326336564096459114688595539499977255560361
int = 71, x = 71.70514887852601938925004856643131493693668265094361525152064731910683
int = 100, x = 100.687957057255903800654040248040422266009951714500281856818692090863
int = 145, x = 145.357909981876032097320383702656272557004669693397154358459436022770
This is pretty much what Hal got above.
~modest
EDIT: and nearly exactly what CraigD reported while I was posting 
