Special Numbers, Special Dates

On the 26th January 2023, I posted about Turning Dates Into Numbers and made the observation that the 10th February 2023 was a special day because the number associated with this date (20230210) had the interesting property that it was the only date in the year that produced a number that had seven distinct prime factors:

20230210 = 2 x 5 x 7 x 11 x 13 x 43 x 47

It turns out that this number has another interesting property in that the sum of its digits equals the last two digits (representing the day of the month). In fact, it begins a run of ten consecutive numbers that end with 20230219 (the 19th February 2023) ... permalink:

20230210, 20230211, 20230212, 20230213, 20230214, 20230215, 20230216, 20230217, 20230218, 20230219

Another run of ten similar numbers occurs in November, starting on the 10th and ending on the 19th.

20231110, 20231111, 20231112, 20231113, 20231114, 20231115, 20231116, 20231117, 20231118, 20231119

For many, many years to come, the sum of the digits of the numbers associated with the dates will always contain two digits. Today is the 15th February 2023 and so we are into the second half of the first-mentioned run.

This number can be expressed as a sum of three cubes
See Figure 2

Similar runs will occur every year but the dates will differ naturally. For example, in 2024, the first run of numbers will begin on 11th January 2024 with 20240111 and end on 19th January 2024 with 20240119. 

In my previous post Turning Dates Into Numbers I looked at what numbers were prime, semiprime and so on. In this post I'll turn my attention to cubes and see what numbers can be expressed as the sum of two cubes and as a sum of three cubes. 

It turns out that there is only one number in the entire year that can be expressed as a sum of two cubes and that is \( 20231020 = 69^3 + 271^3\). This corresponds to the 20th October 2023.

The sums of three cubes are more common. The complete list is shown in Figure 2, arranged in chronological order with 20th January being the first date and 26th December the last.

Figure 2: permalink

It can be noted that some dates produce numbers that can be expressed as sums of three cubes in more than one way. Specifically, 20230426 (two ways), 20230523 (two ways), 20230722 (two ways), 20230904 (two ways), 20230911 (three ways), 20231028 (two ways) and 20231226 (two ways). Thus the 11th September (the famous 911 in United States date format) produces the only number (20230911) that can be expressed as a sum of three cubes in three ways.