Representation of Integers in Golden Mean Number System

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Theorem

The positive integers $n$ are represented in the golden mean number system in their simplest form $S_n$ as follows:

$n$ $S_n$
$1$ $1$
$2$ $10 \cdotp 01$
$3$ $100 \cdotp 01$
$4$ $101 \cdotp 01$
$5$ $1000 \cdotp 1001$
$6$ $1010 \cdotp 0001$
$7$ $10000 \cdotp 0001$
$8$ $10001 \cdotp 0001$
$9$ $10010 \cdotp 0101$
$10$ $10100 \cdotp 0101$
$11$ $10101 \cdotp 0101$
$12$ $100000 \cdotp 101001$
$13$ $100010 \cdotp 001001$
$14$ $100100 \cdotp 001001$
$15$ $100101 \cdotp 001001$
$16$ $101000 \cdotp 100001$


Proof

$1$ is represented by $\left[{1}\right]_\phi = \phi^0$.

From there, the algorithm for Addition of 1 in Golden Mean Number System is run.

\(\text {(2)}: \quad\) \(\ds 1 + \left[{1}\right]_\phi\) \(=\) \(\ds 1 + \left[{0 \cdotp 11}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds \left[{1 \cdotp 11}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{10 \cdotp 01}\right]_\phi\) Simplification


\(\text {(3)}: \quad\) \(\ds 1 + \left[{10 \cdotp 01}\right]_\phi\) \(=\) \(\ds \left[{11 \cdotp 01}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{100 \cdotp 01}\right]_\phi\) Simplification


\(\text {(4)}: \quad\) \(\ds 1 + \left[{100 \cdotp 01}\right]_\phi\) \(=\) \(\ds \left[{101 \cdotp 01}\right]_\phi\)


\(\text {(5)}: \quad\) \(\ds 1 + \left[{101 \cdotp 01}\right]_\phi\) \(=\) \(\ds 1 + \left[{101 \cdotp 0011}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds 1 + \left[{100 \cdotp 1111}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds \left[{101 \cdotp 1111}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{110 \cdotp 0111}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{1000 \cdotp 0111}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{1000 \cdotp 1001}\right]_\phi\) Simplification


\(\text {(6)}: \quad\) \(\ds 1 + \left[{1000 \cdotp 1001}\right]_\phi\) \(=\) \(\ds \left[{1001 \cdotp 1001}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{1010 \cdotp 0001}\right]_\phi\) Simplification


\(\text {(7)}: \quad\) \(\ds 1 + \left[{1010 \cdotp 0001}\right]_\phi\) \(=\) \(\ds \left[{1011 \cdotp 0001}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{1100 \cdotp 0001}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{10000 \cdotp 0001}\right]_\phi\) Simplification


\(\text {(8)}: \quad\) \(\ds 1 + \left[{10000 \cdotp 0001}\right]_\phi\) \(=\) \(\ds \left[{10001 \cdotp 0001}\right]_\phi\)


\(\text {(9)}: \quad\) \(\ds 1 + \left[{10001 \cdotp 0001}\right]_\phi\) \(=\) \(\ds 1 + \left[{10000 \cdotp 1101}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds \left[{10001 \cdotp 1101}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{10010 \cdotp 0101}\right]_\phi\) Simplification


\(\text {(10)}: \quad\) \(\ds 1 + \left[{10010 \cdotp 0101}\right]_\phi\) \(=\) \(\ds \left[{10011 \cdotp 0101}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{10100 \cdotp 0101}\right]_\phi\) Simplification


\(\text {(11)}: \quad\) \(\ds 1 + \left[{10100 \cdotp 0101}\right]_\phi\) \(=\) \(\ds \left[{10101 \cdotp 0101}\right]_\phi\)


\(\text {(12)}: \quad\) \(\ds 1 + \left[{10101 \cdotp 0101}\right]_\phi\) \(=\) \(\ds 1 + \left[{10101 \cdotp 010011}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds 1 + \left[{10101 \cdotp 001111}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds 1 + \left[{10100 \cdotp 111111}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds \left[{10101 \cdotp 111111}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{10110 \cdotp 011111}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{11000 \cdotp 011111}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{100000 \cdotp 011111}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{100000 \cdotp 100111}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{100000 \cdotp 101001}\right]_\phi\) Simplification


\(\text {(13)}: \quad\) \(\ds 1 + \left[{100000 \cdotp 101001}\right]_\phi\) \(=\) \(\ds \left[{100001 \cdotp 101001}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{100010 \cdotp 001001}\right]_\phi\) Simplification


\(\text {(14)}: \quad\) \(\ds 1 + \left[{100010 \cdotp 001001}\right]_\phi\) \(=\) \(\ds \left[{100011 \cdotp 001001}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{100100 \cdotp 001001}\right]_\phi\) Simplification


\(\text {(15)}: \quad\) \(\ds 1 + \left[{100100 \cdotp 001001}\right]_\phi\) \(=\) \(\ds \left[{100101 \cdotp 001001}\right]_\phi\)


\(\text {(16)}: \quad\) \(\ds 1 + \left[{100101 \cdotp 001001}\right]_\phi\) \(=\) \(\ds 1 + \left[{100100 \cdotp 111001}\right]_\phi\) Expansion
\(\ds \) \(=\) \(\ds \left[{100101 \cdotp 111001}\right]_\phi\)
\(\ds \) \(=\) \(\ds \left[{100110 \cdotp 011001}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{101000 \cdotp 011001}\right]_\phi\) Simplification
\(\ds \) \(=\) \(\ds \left[{101000 \cdotp 100001}\right]_\phi\) Simplification

$\blacksquare$


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