Many special functions including the gamma function have a [duplication formula](http://en.wikipedia.org/wiki/Duplication_formula) of some sorts. In the case of the gamma function it reads:



> 
> Gamma(2z) = Gamma(z) Gamma(z+1/2) 22z-1/Gamma(1/2)
> 
> 
> 


On the other hand, there is no algebraic relation between Gamma(2z) and Gamma(z) by themselves, meaning that is there is no nonzero polynomial f(x,y) such that f(Gamma(2z),Gamma(z))=0 for all complex z. I can prove this by chasing poles and their order.


However, I'd be interested in a (simple) argument which shows that the following similar statement is true (which I believe it is):



> 
> There is no (nonzero) polynomial f(x,y) such that f((2n)!, n!)=0 for all integers n≥0.
> 
> 
> 


Any ideas? Thank you!



No simple duplication formula for factorials?
It's equivalent to show that there is no polynomial relationship f({2n choose n}, n!) = 0. On the other hand, we know that {2n choose n} ~ 4^n/sqrt{n} asymptotically and n! grows much faster. 


Terence Tao once remarked that if a sufficiently simple duplication formula were known for the factorial then Wilson's theorem would give an efficient primality test. (Edit: see the other answer. I may be misremembering the stronger remarks that Dick Lipton made.)


