I was born on the 13th. So you'll forgive me if I've never quite understood the fuss.

With another Friday the 13th arriving soon, the number is once again getting its annual dose of suspicion and superstition. While some people avoid the number, hotels skip it on elevator buttons, others dread it on a calendar, I've spent my whole life with it as a kind of companion. I’ve always felt quietly fond of the number 13.

During the height of the Eras Tour, I realized I wasn't alone. Taylor Swift has considered 13 her lucky number for years, literally writing it on her hand before concerts. But 13 is much more than lucky, a mathematical fact I shared with a middle school math circle over a decade ago. The topic of that math circle: why 13 might just be the best number. We spent 90 minutes exploring what it means for a number to be lucky, fortunate, and happy, three distinct ideas in number theory, and 13 qualifies for all three. 

Today I want to share that case with you. Consider it an invitation to follow curiosity wherever it leads, even (or especially!) if it leads straight to the number 13.

13 is a Lucky Number

Not in the everyday sense (though I'd argue that too). In mathematics, a lucky number is defined by a specific filter (or sieve), one of several mathematical processes that systematically eliminate numbers from a list, each isolating a different interesting sequence. This particular sieve, introduced in a 1956 paper by Gardiner, Lazarus, Metropolis, and Ulam, works entirely by position rather than value.

Here's how it works: start with all the positive integers. Cross out every second number, gone are all the even numbers. The next surviving number is 3, so cross out every third number that remains. The next “survivor” tells you what to cross out next, and so on. Keep going, and certain numbers permanently escape elimination. Ulam called them lucky. The sequence begins:

1, 3, 7, 9, 13, 15, 21, 25, 31...

There's 13, still “alive.”

What makes lucky numbers fascinating isn't just the sieve; it's that the lucky numbers and the primes behave remarkably similarly. There are 25 primes less than 100 and 23 lucky numbers less than 100. A version of Goldbach's famous conjecture (that every even number is the sum of two primes) has been extended to lucky numbers, too. Mathematicians aren't entirely sure why two such differently defined sequences share so many properties. But they do.

13 is a Fortunate Number

This one earns its place partly because of who it's named after. 

A Fortunate number is named for Reo Fortune, an anthropologist, interestingly enough, not a mathematician. The motivation was an approach to identify prime numbers (still an unproven conjecture). The definition involves primorials: products of the first n prime numbers. Take such a product, then find the smallest number greater than 1 that you can add to it to get a prime.

So how does 13 earn the title? It comes from the fourth primorial: that is, multiply the first four prime numbers (2 × 3 × 5 × 7 = 210), then find the smallest integer greater than 1 that you can add to 210 to get a prime. It turns out that 13 is exactly that number, since 210 + 13 = 223, which is prime.

The first seven Fortunate numbers, in order, are: 3, 5, 7, 13, 23, 17, 19...

There's 13 again. Fortunate, in every sense of the word.

13 is a Happy Number

You could say that I arrived at happy numbers by turning a frown upside down. I first learned about their opposites when I came across a short article titled "Why Four is the Nemesis of Happy Numbers," in which Esther Inglis-Arkell describes numbers that get “trapped” in an endless loop, never reaching 1. Or as mathematicians might say, never finding their identity. That sent me looking for the happier side of the story.

Here's how it works: take any number. Square its digits and add them together. Take that result and do the same. Keep going. If you eventually reach 1, you have a happy number.

Here's 13:

1² + 3² = 1 + 9 = 10

1² + 0² = 1

Done. Two steps. 13 is happy.

Numbers that aren't happy wander forever; they get trapped in a loop that always passes through 4. Which is, I think, a very sad fate for a number. And perhaps why I've always found 4 a little gloomy (true story!).

(However, there is some small redemption for 4, without it, 13 wouldn't be Fortunate.)

And a Few More Things About 13

You likely know this, but the accolades for 13 keep on coming.

The number 13 is prime. It's the smallest emirp: a prime that gives you a different prime when you reverse its digits. Reverse 13, and you get 31, also prime. It also belongs to a very exclusive club in number theory. (By the way, they just discovered a new largest emirp!)

Thirteen is one of only three known Wilson primes, primes p where (p−1)!+1 is divisible by p2. The only others currently known are 5 and 563. (No one has found a fourth, and they’ve been looking!) 

And 13 is a Fibonacci number, sitting in the sequence right between 8 and 21, part of the mathematical structure behind the spirals in sunflowers and pinecones. It's even responsible for the extra donut or bagel you get as part of a Baker's Dozen!

Not everything is modeling, but it's still about the mindset

Not every encounter with mathematics has to be about building a model. Sometimes it's about following curiosity down a rabbit hole and seeing where it leads.

I agree with Taylor Swift, 13 is definitely lucky! But it's so much more than that, and it's a reminder that curiosity is its own reward. You never know where a number you love might lead you; I just hope it's not under a ladder or involves a black cat.