Today we’re halfway through October, at the measured center of a month that ends with the thinning of the trusty veil holding back the dark from us. Or to take the other side of it, holding the chaos and discord of us back from the slower, calmer, comforting dark. (By the way, remember that word “discord” for a few minutes.)
People have always been afraid of that dark, whatever they call it. The cold of it, the ancient, frozen constancy, holds no vivacity or animation. We may learn early in life that its stony, unchanging stare holds disapproval and disdain. At least we think that’s what we see, although ice is just water caught forever in stillness, and what is still water if not a mirror? But dissemble and rationalize as we might, we know what’s coming.
We know because we count; we incessantly count, and today the counting has reached 10, 15, 20, and 21. Those are numbers that resonate together, suggesting meaning. Maybe that’s because they contain a sequence of fives, and five is an auspicious number in any reckoning. It’s a number most of us encounter each time we use our hands, if nothing else.
Five, as a number, has surprising depth. It’s a prime number, but more than that it’s a prince among primes. First of all, it’s a Fermat pri. That means it’s one of the primes that can be expressed as 2^2^n + 1. Five is the first Fermat prime, so its ’n’ is one.
You may recognize the name “Fermat”, even if you’re not a devotee of mathematics, because Pierre de Fermat left behind a captivating mystery. In 1637 he made a note in the margin of a textbook (Arithmetica, by Diophantus of Alexandria — a book that was over a thousand years old even in Fermat’s day) that he had a proof of an elegant little proof, or theorem, about prime numbers, but it was too large to add in the limited space of the margin. As far as anyone could ever discover — and boy, did they search — he never did write down the proof. But the proof was so enticing, and the notion of finding it was so captivating, that mathematicians tried for centuries to reproduce it. Andrew Wiles finally solved it in 1994, and won the Abel Prize (a sort of Nobel for math) for his accomplishment, which took him six years.
In the intervening years, one of the mathematicians who tried was Sophie Germain. In spite of being born in 1776, when women were not, as a rule, even welcome in schools, she became a physicist, philosopher, and, of course, mathematician. She didn’t solve Fermat’s Last Theorem, but her work was the foundation used for the next couple of centuries. In the process she managed to discover something new about prime numbers, and about the number five, and ever since the number five has not just been a Fermat prime, but a Sophie Germain prime as well.
A number is a Sophie Germain prime if, when you double it and add one, the result is also prime. If you do that to five, you get 11, another prime. But not only is five a Sophie Germain prime, it’s also a “safe prime”. What’s a safe prime, you ask? It’s actually simple. A Sophie Germain prime is a number — call it “n” — that you can put into this formula: 2n+1 — and the result is also a prime. Then if you put that result back into the same formula and get another prime, your original number is a “safe prime.” So 2 x 5 + 1 = 11, which makes 5 a Sophie Germain prime. And if you then calculate 2 x 11 + 1, you get 23, another prime. That makes 5 a safe prime too. What’s so safe about it? I have no idea.
Although this seems like just a parlor game, it’s more than that. Without Sophie Germain primes and safe primes, the cryptography that keeps things like web pages secure would not exist. I could definitely show you how I can prove that, honest I could, but I don’t have the space right here…
The number five appears in a startling range of other functions and results significant enough to have names. Without going into the details, it’s a Catalan number (Eugene Catalan), a Wilson prime (John Wilson), and an Eisenstein prime (Gotthold Eisenstein). It appears in numerical sequences named after Fibonacci, Markov, Pell, and Sierpinski. There’s even something unique about it in the Fibonacci series: that series is infinitely long, but five is the only Fibonacci number that equals its position (fifth) in the sequence.
Many religions include five as an important or holy number in various ways. Even one (probably) fake religion, Discordianism (there’s that word again, “discord”) reveres the “law of fives”, which states “All things happen in fives, or are divisible by or are multiples of five, or are somehow directly or indirectly appropriate to five.” Try not to think too hard about that or you might not be able to keep reading. Regardless, Discordians find it significant that “five” appears exactly five times in that statement. They also explain that in Star Wars, Luke Skywalker’s X-wing fighter is “red five”. I told you it was probably a fake religion.
It doesn’t take advanced math to derive five from the fifteenth day of the tenth month of a year beginning with twenty. The date is just a side effect of the calendar we happen to use, though. There have been countless different calendars across thousands of years and hundreds of civilizations. The one we’re using now is called the Gregorian Calendar, which isn’t all that significant (except possibly to Discordians), but it did begin to be adopted on October 15, 1582. That’s probably worth a high five; it can be a comfort to think that there’s a certain kind of order to everything after all, and discord doesn’t rule the world.