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Let UTM(m, n) be the class of universal Turing machine with m states and n symbols. Universal Turing machines are proved to exist in the following classes: UTM(24,2), UTM(10,3), UTM(7,4), UTM(5,5), UTM(4,6), UTM(3,10) and UTM(2,18).
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Yurii Rogozhin (1996) studied this question.
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