1. http://www.earlham.edu/~peters/courses/logsys/turing2.htm (This proof looks cool. But I think in step (3) it should be $$n+d$$ rather than $$n+1$$.)

2. http://cis.csuohio.edu/~somos/beaver.ps (This proof not only proves $$\Sigma (n)$$, i.e., busy beaver function, is uncomputable, but also shows that $$\Sigma (n)$$ grows faster than any computable function asymptotically.)

3. I think up of a proof using the recursion theorem. Suppose there is a TM $$B$$ that can compute the busy beaver function, then construct a TM $$M$$, starting on blank tape, do the following operation:

• Get its own description $$<M>$$, via the recursion theorem.

• Once you know $$<M>$$, you know the number of states of $$M$$. Denote it by $$k$$. Use $$B$$ to compute the busy beaver function of $$k$$.

• Write $$k+1$$ 1’s on the tape, leading to a contradiction.