tan@ihu1e.UUCP (exit) (05/02/85)
> Now I have a question. > > x^(x^(x^(x^... = 2 implies x = sqrt(2) as shown earlier. > > Then in the equation x^(x^(x^(x^... = 4, we can perform a similar > substitution to obtain > > x^4 = 4 => x = sqrt(2). > > How can the L.H.S. using x = sqrt(2) equal both 2 and 4? > > Jeff Heatwole > ..!hou2e!jlh The answer is that x^(x^(x^...... for x=sqrt(2) is equal to 2 and not to 4. The equation x^(x^(x^..... = 4 has no solution (at least no real solution- I haven't taken the time to check for complex ones). The solution given by Heatwole is erroneous, because the substitution assumes a priori that there is a solution. If there is indeed no solution to the equation x^x^x.... = 4, the substitution of 4 for x^x^x..... is invalid. The same can be said for the "solution" given for x^x^x.... = 2; the substitution only proves that if there is a solution, it must be sqrt(2) (ignoring for the moment the possibility of -sqrt(2)). Actual computation of the limit shows that sqrt(2) here IS a valid solution. -- Bill Tanenbaum AT&T Bell Laboratories - Naperville Ill.