Why does 2 ≠ 4?

Question

Let x ^(x ^(x ^(x^ (...))) = 2
then x ^ 2 = 2 or
x = √2.
So... √2 ^(√2 ^(√2 ^(√2 ^(...))) = 2

But what if z ^(z ^(z ^(z^ (...))) = 4?
then z ^ 4 = 4 or
z = 4^(1/4) = 2^(1/2) = √2 !!
so √2 ^(√2 ^(√2 ^(√2 ^(...))) = 4 !!

Therefore √2 ^(√2 ^(√2 ^(√2 ^(...))) = 2 and
√2 ^(√2 ^(√2 ^(√2 ^(...))) = 4
so 2 = 4.

What's wrong here?

Check it out with a calculator.
what is √2^(√2^(√2^(√2^...)))?
Take it out to 10-12 exponentiations. It'll converge...

6 answers so far and no one is remotely close to a correct analysis.

Note to G.
For x=e^(1/e) , the function converges to e. (2.7182818....)

Second note to G.
There is no error in my reasoning.
let y = x^(x^(x^(x^(...)))) = x^m
then m = x^(x^(x^(x^(...)))) = y !!!
so y = x^y
Therefore if y = 2, x = √2
In other words, the functions:
y = x^(x^(x^(x^(...))))
and
y = x^(1/x)
are inverse functions within the boundaries of their convergence zones.

Answer 1

The function you describe is sometimes known as a power tower, and only converges if 0.0659<=x<=1.4446. if x=sqrt(2), then it converges to 2. For the upper limit, the function converges to somewhere over 2.71 (I can't give you an exact number, I run out of stack space on my machine). But the function can never equal four, it diverges, and equals infinity.
The equality z^(z^(z^((...)))=4 is false. There is no z that can satisfy the equation.
There is a proof by Euler (amongst others). I suggest you look at the reference provided, it'll point you to other resources. I don't think this is trivial at all.

There are other errors in your reasoning.x^2= x*x, not x^x, and z^4=z*z*z*z, not z^z^z^z.
Furthermore, you're choosing an arbitray n at lines 2 and 5, while the equations imply a summation to infinity.

Answer 2

I think that the solution to the paradox is to realise that the two final left-hand terms are not the same thing. Both terms are recursive constructions of even powers of the square root of two. However, the z equation has one more term in it than the x equation.

This is the problem when dealing with infinite recursions. Because the above equations produce multiple results, the paradox is resolved by saying that the recursions are undefined. It's sort of like dividing zero by zero, as since x * 0 = 0, x can be any number.

Nice problem though. I had to think about it.

Answer 3

i think the problem lies in the fact that you have an infinite series. by substituting z as sqrt(2), one term was actually lost hence, leading to the paradox. i think the series √2 ^(√2 ^(√2 ^(√2 ^(...))) converges to 1.

it's similar to a problem i know which proves that 1 < 0. it's one of the many mysteries that an infinite series possesses.

Answer 4

This cannot be correct:

Let x ^(x ^(x ^(x^ (...))) = 2
then x ^ 2 = 2

because x ^(x ^(x ^(x^ (...))) only equals x ^ 2 when x=1. Therefore, neither of the above statements can be true.

Answer 5

series dont converge, i guess for x > 0. so it would be wrong assuming the right hand side

Answer 6

You assume there is an x and z that solves your equations, but these equations have no solutions.

Answer 7

eeh i dont get it... 2 dosnt equl four because two isnt four?