Representing positive integers base 3

Positive whole numbers can easily be written in base 3, there where we allow the digits 0, 1, 2

For example: which we write as , where the notation picks out the digits and puts a subscript “3” to remind us that these digits need to be multiplied by appropriate powers of 3 and then added.

Using this notation and because

Base 3 digits of powers of 2

How do the base 3 representations of powers of 2 look?

Here are the base 3 digits of the first 50 powers of 2:

n	Base 3 digits of
0	1
1	2
2	11
3	22
4	121
5	1012
6	2101
7	11202
8	100111
9	200222
10	1101221
11	2210212
12	12121201
13	102020102
14	211110211
15	1122221122
16	10022220021
17	20122210112
18	111022121001
19	222122012002
20	1222021101011
21	10221112202022
22	21220002111121
23	120210012000012
24	1011120101000101
25	2100010202000202
26	11200021111001111
27	100100112222002222
28	200201002221012221
29	1101102012212102212
30	2202211102201212201
31	12112122212110202102
32	102002022201221111211
33	211011122110220000122
34	1122100021221210001021
35	10021200120220120002112
36	20120101011211010012001
37	111010202100122020101002
38	222021111201021110202011
39	1221120000102112221111022
40	10220010000212002212222121
41	21210020001201012202222012
42	120120110010102102112221101
43	1011010220020211212002212202
44	2022021210111200201012202111
45	11121120121000101102102111222
46	100020011012000202211212000221
47	200110022101001112200201001212
48	1100220121202010002101102010201
49	2201211020111020011202211021102

x

How do the numbers of base 3 digits of vary with ?

Below are plots of the number of occurrences of digits 0, 1, 2 in the base 3 representation of for :

When we plot the running average of the number of occurrences of the digit 2 in the base 3 representation of – that is, the average of the numbers in the third (red) plot above from the very beginning – we get a remarkably straight line:

Erdös conjecture on the digit 2

Paul Erdös noticed that the base 3 representation of contains no 2 digit, but that after that the base 3 representation of powers of 2 seem to always contain the digit 2. He conjectured this is so, and to this day no-one knows whether this is true or not.

If we look again we see that in this list the base 3 representation of contains the digit 2 twice, and that seems to be true for the other powers with

Put another way, seems it might be the last integer for which the base 3 representation of contains just one occurrence of the digit 2, just as seems to be the last integer for which the base 3 representation of contains no occurrences of the digit 2.

We can carry out a search for those integers that are the last integer for which the base 3 representation of contains   or less occurrences of the digit 2.

The search reported below was restricted to .

k	N(k) = largest n such that contains occurrences of the digit 2 ()
0	8
1	24
2	26
3	26
4	36
5	36
6	37
7	42
8	48
9	54
10	76
11	76
12	76
13	76
14	76
15	78
16	110
17	111
18	121
19	134
20	134

Note that .

This means that for  all integers the base 3 representation contains more than 2, and more than 3, occurrences of the digit 2.

As we can see from the first table above, when it seems that the base 3 representation of contains at least 4 occurrences of the digit 2. This remains true in the range .

A plot of the estimates of is shown below:

These are only estimates because we do not know for sure that further calculations for a larger range of might not change these values.

We do not have proofs for any of these estimates.

The occurrence and growth of the digits of in base 3 offer plenty of scope for experimentation by school students and undergraduate students of mathematics.

It seems we can obtain good guesses about the appearance of these digits, without yet knowing in any way how to prove any of these guesses.