Prove that (2011/2012)^(2011/2012) is irrational:

Question

I did try suppose it was a certain arbitrary p/q, but i do not how continue with that proof

Answer 1

Suppose $$\left(\frac{n}{n+1}\right)^{\frac{n}{n+1}}=\frac{a}{b}$$ where $a,b$ have no common factor. Take both sides to the power $n+1$ to get $$\left(\frac{n}{n+1}\right)^n=\left(\frac{a}{b}\right)^{n+1}$$ Clear the denominators: $$n^nb^{n+1}=(n+1)^na^{n+1}$$

Now $a^{n+1}$ and $b^{n+1}$ are coprime, and $n^n$ and $(n+1)^n$ are coprime, so we must have $a^{n+1}=n^n$ and $b^{n+1}=(n+1)^n$. Hence for some $h,k$ we have $a=h^n,n=h^{n+1},b=k^n,n+1=k^{n+1}$. But now $n+1=n+1$ implies $k^{n+1}=h^{n+1}+1$ which is clearly impossible for $n>0$.