Is Ackermann's Function Bijective?

Question

I have been trying to understand the more rigorous side of mathematics and especially functions, and I came across Ackermann's Function recently. I was wondering if A(x,y) is bijective among the natural numbers for any pair (x,y). If it isn't, is it injective or surjective? Any proofs would be greatly appreciated :)

Answer 1

According to wikipedia's table of values: $A(1,3)=5=A(2,1)$.

That disproves injectivity.

For surjectivity just convince yourself that $1$ can't be reached ($2$ can't be either). (As $A(1,n)=n+2$ the function is surjective onto $\mathbb N\setminus\{1,2\}$).