Keith Number/Examples/129,572,008

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Examples of Keith Number

$129 \, 572 \, 008$ is a Keith number:

$1, 2, 9, 5, 7, 2, 0, 0, 8, \ldots, 32 \, 456 \, 930, 64 \, 849 \, 899, 129 \, 572 \, 008, \ldots$


Proof

By definition of Keith number, we create a Fibonacci-like sequence $K$ from $\left({1, 2, 9, 5, 7, 2, 0, 0, 8}\right)$:

\(\ds K_0\) \(=\) \(\ds 1\)
\(\ds K_1\) \(=\) \(\ds 2\)
\(\ds K_2\) \(=\) \(\ds 9\)
\(\ds K_3\) \(=\) \(\ds 5\)
\(\ds K_4\) \(=\) \(\ds 7\)
\(\ds K_5\) \(=\) \(\ds 2\)
\(\ds K_6\) \(=\) \(\ds 0\)
\(\ds K_7\) \(=\) \(\ds 0\)
\(\ds K_8\) \(=\) \(\ds 8\)
\(\ds K_9\) \(=\) \(\ds K_0 + K_1 + K_2 + K_3 + K_4 + K_5 + K_6 + K_7 + K_8\)
\(\ds \) \(=\) \(\ds 1 + 2 + 9 + 5 + 7 + 2 + 0 + 0 + 8\)
\(\ds \) \(=\) \(\ds 34\)
\(\ds K_{10}\) \(=\) \(\ds K_1 + K_2 + K_3 + K_4 + K_5 + K_6 + K_7 + K_8 + K_9\)
\(\ds \) \(=\) \(\ds 2 + 9 + 5 + 7 + 2 + 0 + 0 + 8 + 34\)
\(\ds \) \(=\) \(\ds 67\)
\(\ds K_{11}\) \(=\) \(\ds K_2 + K_3 + K_4 + K_5 + K_6 + K_7 + K_8 + K_9 + K_{10}\)
\(\ds \) \(=\) \(\ds 9 + 5 + 7 + 2 + 0 + 0 + 8 + 34 + 67\)
\(\ds \) \(=\) \(\ds 132\)
\(\ds K_{12}\) \(=\) \(\ds K_3 + K_4 + K_5 + K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11}\)
\(\ds \) \(=\) \(\ds 5 + 7 + 2 + 0 + 0 + 8 + 34 + 67 + 132\)
\(\ds \) \(=\) \(\ds 255\)
\(\ds K_{13}\) \(=\) \(\ds K_4 + K_5 + K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12}\)
\(\ds \) \(=\) \(\ds 7 + 2 + 0 + 0 + 8 + 34 + 67 + 132 + 255\)
\(\ds \) \(=\) \(\ds 505\)
\(\ds K_{14}\) \(=\) \(\ds K_5 + K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13}\)
\(\ds \) \(=\) \(\ds 2 + 0 + 0 + 8 + 34 + 67 + 132 + 255 + 505\)
\(\ds \) \(=\) \(\ds 1003\)
\(\ds K_{15}\) \(=\) \(\ds K_6 + K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14}\)
\(\ds \) \(=\) \(\ds 0 + 0 + 8 + 34 + 67 + 132 + 255 + 505 + 1003\)
\(\ds \) \(=\) \(\ds 2004\)
\(\ds K_{16}\) \(=\) \(\ds K_7 + K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15}\)
\(\ds \) \(=\) \(\ds 0 + 8 + 34 + 67 + 132 + 255 + 505 + 1003 + 2004\)
\(\ds \) \(=\) \(\ds 4008\)
\(\ds K_{17}\) \(=\) \(\ds K_8 + K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16}\)
\(\ds \) \(=\) \(\ds 8 + 34 + 67 + 132 + 255 + 505 + 1003 + 2004 + 4008\)
\(\ds \) \(=\) \(\ds 8016\)
\(\ds K_{18}\) \(=\) \(\ds K_9 + K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17}\)
\(\ds \) \(=\) \(\ds 34 + 67 + 132 + 255 + 505 + 1003 + 2004 + 4008 + 8016\)
\(\ds \) \(=\) \(\ds 16 \, 024\)
\(\ds K_{19}\) \(=\) \(\ds K_{10} + K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18}\)
\(\ds \) \(=\) \(\ds 67 + 132 + 255 + 505 + 1003 + 2004 + 4008 + 8016 + 16 \, 024\)
\(\ds \) \(=\) \(\ds 32 \, 014\)
\(\ds K_{20}\) \(=\) \(\ds K_{11} + K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19}\)
\(\ds \) \(=\) \(\ds 132 + 255 + 505 + 1003 + 2004 + 4008 + 8016 + 16 \, 024 + 32 \, 014\)
\(\ds \) \(=\) \(\ds 63 \, 961\)
\(\ds K_{21}\) \(=\) \(\ds K_{12} + K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20}\)
\(\ds \) \(=\) \(\ds 255 + 505 + 1003 + 2004 + 4008 + 8016 + 16 \, 024 + 32 \, 014 + 63 \, 961\)
\(\ds \) \(=\) \(\ds 127 \, 790\)
\(\ds K_{22}\) \(=\) \(\ds K_{13} + K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21}\)
\(\ds \) \(=\) \(\ds 505 + 1003 + 2004 + 4008 + 8016 + 16 \, 024 + 32 \, 014 + 63 \, 961 + 127 \, 790\)
\(\ds \) \(=\) \(\ds 255 \, 325\)
\(\ds K_{23}\) \(=\) \(\ds K_{14} + K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22}\)
\(\ds \) \(=\) \(\ds 1003 + 2004 + 4008 + 8016 + 16 \, 024 + 32 \, 014 + 63 \, 961 + 127 \, 790 + 255 \, 325\)
\(\ds \) \(=\) \(\ds 510 \, 145\)
\(\ds K_{24}\) \(=\) \(\ds K_{15} + K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23}\)
\(\ds \) \(=\) \(\ds 2004 + 4008 + 8016 + 16 \, 024 + 32 \, 014 + 63 \, 961 + 127 \, 790 + 255 \, 325 + 510 \, 145\)
\(\ds \) \(=\) \(\ds 1 \, 019 \, 287\)
\(\ds K_{25}\) \(=\) \(\ds K_{16} + K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24}\)
\(\ds \) \(=\) \(\ds 4008 + 8016 + 16 \, 024 + 32 \, 014 + 63 \, 961 + 127 \, 790 + 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287\)
\(\ds \) \(=\) \(\ds 2 \, 036 \, 570\)
\(\ds K_{26}\) \(=\) \(\ds K_{17} + K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25}\)
\(\ds \) \(=\) \(\ds 8016 + 16 \, 024 + 32 \, 014 + 63 \, 961 + 127 \, 790 + 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287 + 2 \, 036 \, 570\)
\(\ds \) \(=\) \(\ds 4 \, 069 \, 132\)
\(\ds K_{27}\) \(=\) \(\ds K_{18} + K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26}\)
\(\ds \) \(=\) \(\ds 16 \, 024 + 32 \, 014 + 63 \, 961 + 127 \, 790 + 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287 + 2 \, 036 \, 570 + 4 \, 069 \, 132\)
\(\ds \) \(=\) \(\ds 8 \, 130 \, 248\)
\(\ds K_{28}\) \(=\) \(\ds K_{19} + K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27}\)
\(\ds \) \(=\) \(\ds 32 \, 014 + 63 \, 961 + 127 \, 790 + 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287 + 2 \, 036 \, 570 + 4 \, 069 \, 132 + 8 \, 130 \, 248\)
\(\ds \) \(=\) \(\ds 16 \, 244 \, 472\)
\(\ds K_{29}\) \(=\) \(\ds K_{20} + K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28}\)
\(\ds \) \(=\) \(\ds 63 \, 961 + 127 \, 790 + 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287 + 2 \, 036 \, 570 + 4 \, 069 \, 132 + 8 \, 130 \, 248 + 16 \, 244 \, 472\)
\(\ds \) \(=\) \(\ds 32 \, 456 \, 930\)
\(\ds K_{30}\) \(=\) \(\ds K_{21} + K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28} + K_{29}\)
\(\ds \) \(=\) \(\ds 127 \, 790 + 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287 + 2 \, 036 \, 570 + 4 \, 069 \, 132 + 8 \, 130 \, 248 + 16 \, 244 \, 472 + 32 \, 456 \, 930\)
\(\ds \) \(=\) \(\ds 64 \, 849 \, 899\)
\(\ds K_{31}\) \(=\) \(\ds K_{22} + K_{23} + K_{24} + K_{25} + K_{26} + K_{27} + K_{28} + K_{29} + K_{30}\)
\(\ds \) \(=\) \(\ds 255 \, 325 + 510 \, 145 + 1 \, 018 \, 287 + 2 \, 036 \, 570 + 4 \, 069 \, 132 + 8 \, 130 \, 248 + 16 \, 244 \, 472 + 32 \, 456 \, 930 + 64 \, 849 \, 899\)
\(\ds \) \(=\) \(\ds 129 \, 572 \, 008\)


Thus $129 \, 572 \, 008$ occurs in $K$ and the result follows by definition of Keith number.

$\blacksquare$


Historical Note

The Keith number $129 \, 572 \, 008$ was discovered, along with $251 \, 233 \, 297$, by Clifford A. Pickover, who reported it in his work Computers and the Imagination of $1991$.


Sources

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