(Bloomberg Opinion) -- The year 2019 capped off a decade in which some of the thorniest math questions finally yielded to mathematicians' ingenuity. We learned profound facts about the distribution of primes, approximations of irrational numbers and how to pack eight- and 24-dimensional oranges (not so useful for grocers, but important for communications technology).
But math geeks like me don't just get excited about the breakthroughs in recent years – we're also delighted by the mathematical properties of those years themselves. And by that metric, the last decade was spectacular: Indeed, it contained two prime-numbered years — 2,011 and 2,017 have no divisors other than 1 and themselves. There was also one year (2016) that was icositetragonal, meaning that it can be represented by a set of pebbles in the shape of a regular 24-sided polygon.
Meanwhile, when you write 2,015 in binary, you get 11111011111 — a palindrome containing just a single "0," which looks a bit like the eye of Sauron. We're unlikely to be around for the next year with that property, which is 8127 (in binary: 1111110111111).
We saw palindrome weeks every year from 2011 through 2019, where the (U.S.-formatted) dates read the same forward and backward. The last decade also saw the only dates this century lining up perfectly with the mathematical constants π (a circle's circumference divided by its diameter) and e (the base of the natural logarithm, which shows up everywhere from calculus to compound interest), as well as the Golden ratio (a proportion found in art, architecture and nature).
And 2019 itself was numerically exciting. As I wrote last year, 2,019 is happy-go-lucky, signifying that it has two intriguing mathematical properties: It is happy, which means that if you take the sum of the squares of 2,019’s digits, and then iteratively take the sums of the squares of the digits of the numbers that result, you eventually reach 1. And 2,019 is lucky, in that it survives a numerical elimination process with mysterious ties to the primes. This was the last happy-go-lucky year for almost a century – the next one is 2115. On top of that, every number that divides 2,019 is lucky, as well.
The number 2,019 also has its own ties to π: it’s the first four-digit number to appear six times in π’s decimal expansion.
This past year was in many ways a defining one for the U.S. and, as it happens, it shares mathematical kinship with one of the most significant years in the nation's history: 2,019 = 1 + 4 + 8 + 9 + ... + 243 is the sum of the first 22 perfect powers. And the sum of the first 21 perfect powers is 1,776.
And 2,019 is also an apocalyptic power, meaning that if we raise 2 to the 2,019th power, we get a figure that contains 666, the number of the beast. (That certainly shouldn’t be taken as an explanation for any recent events — but if you’re the superstitious type, then I’m sorry to tell you that another apocalyptic power is just two years away. )
With all that mathematical excitement behind us, can 2020 live up? Well, 2,020 is an example of what mathematicians call an autobiographical number — so I guess we'll have to wait and let next year tell its own story. (And hindsight of course, is 20-20.)
We also learned how to write 33 as a sum of three cubes of integers – which is surprisingly difficult, requiring cubes of numbers in the quadrillions.
The next icositetragonal number is 2325.
Apply for that zombie apocalypse scholarship ASAP!
An autobiographical number is one in which the Nth digit (reading from the left) tells us the number of times that the digit N appears in the number. So 2,020 is autobiographical because it is made up of two zeros, zero ones, two twos and zero threes. These numbers are extremely 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.
To contact the editor responsible for this story: James Greiff at jgreiff@bloomberg.net
This column does not necessarily reflect the opinion of the editorial board or Bloomberg LP and its owners.
Scott Duke Kominers is the MBA Class of 1960 Associate Professor of Business Administration at Harvard Business School, and a faculty affiliate of the Harvard Department of Economics. Previously, he was a junior fellow at the Harvard Society of Fellows and the inaugural research scholar at the Becker Friedman Institute for Research in Economics at the University of Chicago.
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