User blog:Alemagno12/Calculating values of A(x,y) mod z

Inspired by this blog post (in the japanese GWiki).

When I say that f(x) = a0,a1,a2,...,an,{b0,b1,b2,...,bm} for some function f (like f(x) = A(2,x) mod 3), I mean that f(k) = ak for k ≤ n and that after that the values of f(k) loop through the bps.

Calculating A(x,y) mod 2^z
By definition A(0,x) = x+1, and it can be easily checked that A(1,x) = x+2, so: The extremal case would be z = 1 - since A(0,y) is always odd and A(1,y) is always even, A(0,y) mod 2 = {1,0} and A(1,y) mod 2 = {0,1}.
 * A(0,y) mod 2z = {1,2,3,4,...,2z-2,2z-1,0}, and
 * A(1,y) mod 2z = {2,3,4,5,...,2z-2,2z-1,0,1}.

Next, we have that A(2,x) = 2x+3 - for x = 0, A(2,0) = A(1,1) = 1+2 = 3, and if A(2,n) = 2n+3, then A(2,n+1) = A(1,A(2,n)) = A(2,n)+2 = 2n+3+2 = 2(n+1)+3. So: The extremal case would be z = 1 - since 2x is always even and 3 is odd, 2x+3 = A(2,x) is always odd, so A(2,x) mod 2 = {1}.
 * A(2,y) mod 2z = {3,5,7,9,...,2z-3,2z-1,1}.

[WIP]