Saturday, February 12, 2022

2022, for example

As we entered the current calendar year, James Propp (in a math-fun forum) noted a Dan McQuillan tweet on the embedding of the integer 2022 in its base-three representation:

2202220

The property is not the sole domain of 2022, but rather of a sequence of such integers wherein 2022 is the sixteenth term. James wanted to find out if this sequence was infinite. His argument had it that for length-d integers there should be slightly more than d solutions. Actual number of solutions for d up to 26 are:

3, 3, 6, 4, 4, 0, 0, 3, 3, 10, 0, 15, 14, 20, 8, 13, 20, 25, 9, 21, 14, 20, 23, 17, 8, 18.

That's 281 solutions altogether, thus far. Here are the last 18 (26-digit) solutions:

12011102001112020010002201020010222122111111112022112
12011102001112020010010002110100202102022110210022001
12120120221001100110120020101011000001200201100221210
12120200122001222012202121222002101201220202020112112
12121211002200121012200100101222020222010202110221012
12121211010102002200002122022001122122022111002211220
12121211102211101101221101122010121002020112121202201
12121211110112212210222021122222211212101012002120022
12121211202222011121002000222100101122011102212211110
12121211210200212010210010102210210002222221002210221
12121221220001002121001110110111221111010011221120222
12121222012222220211020021122201211121212110012112122
12121222020000010000010220110120001111001100122121202
12200002021122110111202210110110201200122021112110022
100202112102120012102220200111102002122102011100001122
100202112102120102102210221000121021121102121111011222
100211210200212022120220202012210222011001201111010120
100211210200212022120221010210222021011222010102002111

[last updated 28 March 2022]