bobl (04/29/83)
To solve the problem
...
x
x
x = 2
...
x
note that the exponent of x is x itself, which we claim is 2,
so the equation becomes
2
x = 2
so x = (+ or -) sqrt(2). (High school math WAS useful for something!)
- Bob Lewis
...tektronix!iddic!tekgds!boblucbcad:moore (04/30/83)
#R:ucbvax:-48000:ucbcad:11900001:000:717
ucbcad!moore Apr 29 02:55:00 1983
...
x
x
x = 2.
What is the value of x?
since I don't have a picture mode editor, I will restate this as
x**x**x**x**..... = 2.
if we assume that x**x**x = x**(x**x) (right to left associative)
then x**x**x... = x**(x**x**x...) = x**2 => x**2 = 2. => x = sqrt(2).
We have implicitly assumed that an answer exists here, in that we
are manipulating x**x**x... as though it was a well-behaved quantity.
As an example of the pitfalls, the problem x**x**x.. = 4 has the same
solution, i.e. x = sqrt(2)! Does anyone know what the largest n is
such that x**x**x.. = n has a solution? I suspect e, but have yet
to prove it.
Peter Moore
moore@Berkeley
...!ucbvax!ucbcad!moore halle1 (05/11/83)
x**x**x**...=n has a solution for all n >= 0. (n=0 is an asymptotic
solution, i.e. lim n->0)
>From the earlier article, x**x**x**x...=x**(x**x**...)=n ==> x**n=n.
Taking logs: n ln(x)= ln(n) or ln(x)= ln(n) /n
Therefore: x=exp( ln(n) /n) = (exp( ln(n))**(1/n)
=> x=n**(1/n)
1/n
x = n is defined for all n>o, equals 0 at n=0, and is
undefined (in the real plane) for n<0. (Special cases excepted.)
The maximum value of x occurs at n=e (2.71828182859....) x=1.4447 (rounded).