# Ackerman function

**Algebra**Level 4

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

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

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

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

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

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