7.3: Unusual Number systems

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Dualtown Number System

The peoples of Dualtown use only numbers which are 1 and some of the multiples of 2 (even numbers). \(E=\{1, 2, 4, 6,8 \cdots\}\). Notice that the set \(E\) is closed under multiplication.

Let's construct a \( 4 \times 4\) multiplication table \(E\).

\(\times\)	\(1\)	\(2\)	\(4\)	\(6\)
\(1\)	\(1\)	\(2\)	\(4\)	\(6\)
\(2\)	\(2\)	\(4\)	\(8\)	\(12\)
\(4\)	\(4\)	\(8\)	\(16\)	\(24\)
\(6\)	\(6\)	\(12\)	\(24\)	\(36\)

The smallest ten prime numbers in Dualtown are \(2, 6, 10, 14, 18, 22, 26, 30, 34, 38\). Notice that \(36=(6)(6)=(2)(18)\). Thus \(36\) has two different Dualtown prime factorizations. Hence the Prime Divisibility Theorem does not hold for the Dualtown number system.

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