Ackerman function

Algebra Level 4

Ackermann(–Péter) function AA is a fairly well-known function that grows rapidly. It is defined as follows:

A(m,n)={n+1if m=0A(m1,1)if m>0,n=0A(m1,A(m,n1))if m>0,n>0A(m,n) = \begin{cases} n+1 & \text{if } m = 0 \\ A(m-1, 1) & \text{if } m > 0, n = 0 \\ A(m-1, A(m, n-1)) & \text{if } m > 0, n > 0 \end{cases}

Define Ackerman function A(m,n)\mathcal{A}(m,n) as follows:

A(m,n)={n+1if m=0A(m1,1)if m>0,n=0A(m1,A(m1,n1))if m>0,n>0\mathcal{A}(m,n) = \begin{cases} n+1 & \text{if } m = 0 \\ \mathcal{A}(m-1, 1) & \text{if } m > 0, n = 0 \\ \mathcal{A}(m-1, \mathcal{A}(m-1, n-1)) & \text{if } m > 0, n > 0 \end{cases}

Compute the last 3 digits of i=010j=010A(i,j)\displaystyle\sum_{i=0}^{10} \sum_{j=0}^{10} \mathcal{A}(i,j).

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