TITLE: Is $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ algebraic or transcendental?
QUESTION [84 upvotes]: I thought it was easy to show that $\sqrt {2 \sqrt {3 \sqrt {4 \ldots}}}$ is irrational, but found a gap in my proof. Simple finite approximations show the denominator cannot be small, though, strongly suggesting irrationality. However, can it be shown whether this number is algebraic or transcendental? My hunch is that it's transcendental but I have no idea on how to start such a proof (especially since I can't fill my gap to prove irrationality).

REPLY [4 votes]: There exists a finite difference formulation of this problem. This doesn't solve your problem, but it may be at least a bit enlightening, Consider $f(x)$ such that 
$$ f(x) = x\sqrt{f(x+1)} $$ 
Then it follows 
$$ f(x) = x \sqrt{(x+1) \sqrt{(x+2) \sqrt{... }}} $$ 
And $f(1)$ is the value you wish to evaluate.  
Another way to write that is 
$$ \frac{f(x)^2}{x^2} = f(x+1) $$ 
Let $f(x) = 2^{g(x)} $
$$ \frac{2^{2g(x)}}{x^2} = 2^{g(x+1} $$ 
Taking the logarithm of both sides base-2
$$ 2g(x) - 2\log_2{x} = g(x+1) $$ 
We can now evaluate, this as a run of the mill finite difference equation in G
$$ D_{1,x}g - g = -2\log_2{x} $$ 
(... lots of work that you can comment and request if you want ...)
$$ 2^{1-x}g = -D_{1,x}^{-1}\left[2^{1-x}\log_2{x} \right] $$ 
So it follows that 
$$ g = -2^{x-1}D_{1,x}^{-1}\left[2^{1-x}\log_2{x} \right] $$ 
And that 
$$ f = 2^{ -2^{x-1}D_{1,x}^{-1}\left[2^{1-x}\log_2{x} \right]} $$ 
Note that $f(0) = 0$ and therefore the expression
$$ D_{1,x}^{-1}\left[2^{1-x}\log_2{x} \right] $$ 
must tend to positive infinity as $x$ tends to 0. Analytically i'm not sure how to continue this. But putting it into Wolfram Alpha yields:
http://www.wolframalpha.com/input/?i=f%28x%2B1%29+-+f%28x%29+%3D+2%5E%7B1-x%7Dlog_2%28x%29
That the expression is dependent on polylogarithms (unsurprising) and something called the lerch transcendent (I believe these have to do with generalizing the riemann zeta function). 
So if you choose to go down the finite difference path, it seems that you might be using some heavy analytic number theory tools to determine if the expression is transcendental.