## Archive for June, 2013

### tetration

Here is a problem from fakalin. Find the error in the following proof:

Let $$x^{x^{x\cdots}} = 2$$. Then $$x^{x^{x\cdots}} = x^{x^{x^{x\cdots}}} = 2$$. Substituting, we get $$x^2=2$$ and $$x=\sqrt{2}$$. Now let $$x^{x^{x\cdots}} = 4$$. Similarly, $$x^{x^{x^{x\cdots}}} = 4$$, $$x^4=4$$ and $$x=\sqrt{2}$$. But then $$2=x^{x^{x\cdots}}=4$$ and $$2=4$$.