Divisible only by itself and one. Remarkably simple, but simply amazing.

“Prime, any positive integer greater than 1 that is divisible only by itself and 1” [⇣]. For years I've associated primes with this definition and didn't think much of it. Primes were a mere abstraction; their profundity obscured up until some recent idle thoughts.

“A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.” [⇣]. This definition better conveys their extraordinary nature, but it's still no match for actually visualizing what “indivisible” or “not a product” means.

996 = 1 × 996 = 2 × 498 = 3 × 332 = 4 × 249 = 6 × 166 = 12 × 83 = 2 × 2 × 3 × 83. Simply put, it is possible to represent a collection of 996 things as multiple equal collections of a smaller size. For instance, below is 996 represented as 3 collections of 332 letters “i”.

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{iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii}.

Okay, so what?

997 = 1 × 997. That's it. Merely adding 1 to 996 makes a collection of 997 things, a collection that is *impossible* to represent as any other amount of equal collections of a smaller size—“impossible”!

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What the heck? It goes to show how some *one* can make a difference—how *you* can change the world. Remarkable.

Now consider the prime 67 280 421 310 721, which reads “67 trillion 280 billion 421 million 310 thousand 721”. A number that big surely must be divisible? Nope. Let's go even bigger. The current largest known (Mersenne) prime is:

2^{82589933} - 1 = 148894445742041325547…(24 862 006 digits omitted)…037951210325217902591.

So, primes can be unfathomably large, are infinite in supply [⇣], yet remain indivisible. Seriously, how?

- 2023/01/22: 1. Adds definitions. 2. Adds a sentence. 3. Changes title.

Ψ

- Britannica: prime.
- Wikipedia: prime number.
- The University of Utah: Why are there infinitely many prime numbers?

- Wikipedia: Mersenne prime.
- Great Internet Mersenne Prime Search: Why join GIMPS?
- Vicky Neale: Closing the Gap: The Quest to Understand Prime Numbers.

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