They say that hindsight is 20/20, and like many, I can’t wait for 2020 to become hindsight. But for math geeks like me, one thing was sure in a year of so much uncertainty — admittedly something far abstracted from everything that was going on in the world: 2020 was one of the most numerically exciting years in a long time.
The year featured a number of mathematical holidays: Feb. 2 was palindrome-ambigram-Groundhog Day: Its date, 02/02/2020, was a palindrome (meaning it reads the same forward and backward) and an ambigram (it reads the same if you rotate it 180 degrees). This works in month-first, day-first and year-first date conventions.
There will be another palindrome-ambigram day next year, at least under the month-first convention (12/02/2021). And under a slightly different date convention, a palindrome-ambigram day falls on Groundhog Day again in 2022 (i.e., on 2/2/22). But after that, the next palindrome-ambigram-Groundhog Day won’t be for a century — it’s in 2122.
And Dec. 16 (12/16/20) was a Pythagorean triple day: The month, date and (shortened) year together represented the sides of an integer right triangle, meaning that the sum of the squares of the month and the day is equal to the square of the year: 12^2 + 16^2 = 144 + 256 = 400 = 20^2. That’s the trifecta for math squares everywhere — and we won’t have another one until July 24, 2025.
Even July 13, 2020, was a mathematical holiday — albeit a more obscure one. In the month-first convention (07/13/20), it shows up as consecutive elements in a mysterious mathematical sequence introduced by Recamán that is constructed as follows: You start with 0 and then for each positive integer N, the N-th element of the sequence is equal to the (N-1)st element minus N if that is a positive number not already in the sequence; otherwise, the N-th element is equal to the (N-1)st element plus N. Sound wild? The sequence is even more surreal when put to music — check out several audio renditions here.
The year 2020 is numerically special as well: It’s the only year for just more than a century with double-digit Arabic numerals (the next one is 2121). Plus 2,020 has a remarkably short Roman numeral representation (MMXX), again with repeated pairs.
But there’s far more to 2,020 than that.
As I pointed out last year, 2,020 is an example of what mathematicians call an autobiographical number: Its first digit indicates how many of its digits are 0s (there are two); its second digit indicates how many digits are 1s (there are zero); and so forth. These numbers are exceedingly rare — the only ones are 1,210, 2,020, 21,200, 3,211,000, 42,101,000, 521,001,000, and 6,210,001,000. So we won’t see another one this decamillennium.
The number 2,020 also has a beautiful representation as 202 + 202 + 202 + 202 + 202 + 202 + 202 + 202 + 202 + 202. And it can be written as the sum of the squares of four consecutive primes (2,020 = 17^2 + 19^2 + 23^2 + 29^2); the next year with that property is 2692.
Plus 2,020 is divisible by its reversal (which doesn’t happen again until 2100), and 2,020’s sum of digits equals its number of digits (which next happens in 2101). And 2,020 is “self-slideable” in the sense that you get the same number if you slide each digit d by d places in either direction.
These numerical delights don’t make up for all the turmoil and loss of the past year, of course. But they do give us one opportunity for positive reflection — we might even say, positive integer reflection — as we transition into 2021. And of course (2020)^1024 + (2021)^1024 is a prime number, albeit one slightly too large to print in the text of this column 4 — so there’s hope that next year will bring us at least a bit more prime time.
(Bloomberg)
