On the Non-existance of Rotation-Symmetric von Neumann Neighbor Number-Conserving Cellular Automata of Which the State Number is Less than Four

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A number-conserving cellular automaton (NCCA) is a cellular automaton such that all states of cells are represented by integers and the total number of its configuration is conserved throughout its computing process. In constrast to normal cellular automata, there are infinitely many assignments of states for NCCAs with a constant state number. As for von Neumann neighbor(radius one) NCCAs with rotation-symmetry, a local function can be represented by summation of four binary functions. In this paper, we show that the minimum size of state set of rotation-symmetric von Neumann neighbor NCCA is 5 by using this representation.