By repeatedly replacing the value of x with the formula on the left side, x converges rapidly. Starting with a guess of x=2, it converges thus:
2.300761708156168
2.302761656377654
2.302775540639462
2.302775637057741
2.302775637727312
2.302775637731962
2.302775637731994
2.302775637731995
2.302775637731995
...
I squared the terminal number to see if this was the square root of an integer or of a rational number, which it was not, but it was exactly 3 higher than the number itself (it came out to 5.302775637731996, the difference in the final digit attributable to rounding differences).
So, solving x^23 = x:
x^2  x  3 = 0
x = (1 +/ sqrt(1 + 12)) / 2
As the number is positive, we get (1 + sqrt(13)) / 2.
The program:
DEFDBL AZ
x = 2
FOR i = 1 TO 40
x = SQR(4 + SQR(4  SQR(4 + SQR(4  x))))
PRINT x
NEXT

Posted by Charlie
on 20041224 15:10:29 