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I am looking the way to proceed further in following difficulties. Observe the following: \begin{align} n &= 3 + 3(6) + 3[6^2] + 3[6^3] & \text{$(1)$} \\ &= 129 \end{align} Now by writing $129$ in base $6$, we see the repeated digit $3$ up to three times.

If we took one more term $3[6^3]$ in $(1)$, we will get repeated $3$ up to $4$ times and so on.

Now, my question is, how it is repeated and why this happening like this? is there any reason beside the adding number of terms and having those many threes in base $6$.

If we replace $3$ by $a$ and $6$ by $k$, for writing $n$ in base $k$, we get repeated a up to $m$ times, where $m$ is number of terms in $(1)$.

Please answer...

  • 0
    It is hard to understand your question, Can you please elaborate? Why what is "happening like this"?2012-05-24
  • 1
    This seems to be quite general and to follow directly from the very definition of the decomposition of integers in a given base.2012-05-24
  • 1
    What is the connection to perfect numbers?2012-05-24
  • 2
    I have voted to close. The question is basically asking why $aaa \dots aaa$ is a sequence of $a$'s in base $b$. Answer: because you constructed it that way.2012-05-24

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