## Digits of Powers of 2 in Ternary Numeral System by Yagub Aliyev

Interesting patterns in the ternary digits of powers of two

Let us write first powers of two (1,2,4, , 218) in ternary numeral system so that their digits in the
corresponding place values are aligned along vertical columns.
We can easily observe some interesting patterns in this short table. There are blocks of 0’s and 2’s in the
shape of stairs, larger ones of which seem to have sizes which increase unboundedly. This means that
there are arbitrarily large such stairs. Each stair of the stairs is of height either 1 or 2 digits. The base 3
representation of the above numbers were calculated using Maple. Example:

```> convert(2^31, base, 3);
[2, 0, 1, 2, 0, 2, 0, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1]```

The topic has many interesting connections with The Euler Phi-Function, primitive roots, Benford's law,
Erdőss conjecture about the powers of 2 which have ternary expansions that omit the digit 2, etc.

The picture was created by me (numbers and stairs) and my daughter Khadija (11 yrs old)
(coloring) for Maple Art and Creative Works Exhibit

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