A Digit Shifted, Sums of Digits Sequence

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A Digit Shifted, Sums of Digits Sequence

• Shauna Knarr
• Richard Brazier
• Penn State DuBois

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Base 10

• 918 183 123 63 92 112 43 72 92
• (918)103183
• 919 193 133 73 102 33 62 82 102
• 2 cycles of length 4 , 2-3 digits long are these
• the only ones?

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Base 7

• 666 243 123 63 122 53 112 43 102 33 62 112
• (666)73 243 (18 in base 7 is 24)
• 1 cycle of length 5, 2-3 digits

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Overview

• Introduce Sequence
• Show in base 10 a 3 digit number gets smaller
• General proof for base b, k digits
• Base 10 has 2 cycles of length 4
• Base 7 has 1 cycle of length 5
• What has been proved and what has not been proved

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What is left to prove?

• For base 3 , we can show 5 digits gets smaller,
  and 3 digits get bigger, but 4 digits?? If it
  does cycle at 4 digits it is a fixed point.
• For base 2 it is trapped between 5 and 6 digits
  but there is no proof that it cycles!