Are we including it just to poke the bear? Prime numbers are typically defined as numbers with no positive divisors other than one and the number in question.
Well, I’m questioning it… not that it really matters.
-1, 0, and 1 are no good.
Two divisors for positive integer X:
1, X
I recently demonstrated GPT-4 via by having it explain phases of matter metaphorically with animals. Though impressive, it points to loose abstraction being more in range than firm abstraction. Let’s get that model a python interpreter and see how far we can take this party. Until then, I’m going to stick to explaining concepts.
————- included here for fun
Alright! Let's dive into the wild world of matter using some animal friends as examples:
1. *Solids* - Think of solids like a herd of elephants standing closely together. The elephants are packed in tight, barely moving, just maybe swaying a bit. They're sturdy and strong, just like solid things in our world. This is because the particles in a solid don't move around much; they just vibrate in place.
2. *Liquids* - Imagine a school of fish swimming in a pond. They're free to move around, weaving in and out, but they still stay pretty close to each other. They aren't packed as tightly as our elephant friends, but they aren't completely free either. This is like liquids: the particles are close, but they can move around and flow, just like water in a glass.
3. *Gases* - Now, picture a flock of birds soaring in the sky, free to fly in all directions. These birds aren't sticking close to one another; they're spread out, enjoying the vast space of the sky. In gases, the particles are like these birds, very spread out and moving all over the place.
4. *Plasma* - Think of plasma like dragons (I know they're mythical, but bear with me). These dragons breathe fire, and that fire is so hot and energetic that it can change the way things behave. Plasma is like that – it's gas that's become so hot that the particles are super energized and can even glow, like neon lights.
5. *Bose-Einstein Condensate (BEC)* - This one's a bit trickier, but imagine penguins in Antarctica. They huddle closely together to keep warm. BEC is like the coldest group of particles ever, where they start acting in strange, uniform ways, almost like one giant super-particle. It's like all the penguins moving together as one.
So, next time you think of matter, just remember our animal friends: the sturdy elephants, the flowing fish, the free-flying birds, the fiery dragons, and the huddled penguins!
Again, 1 fits, because it has two divisors: 1, and 1. You never said X != 1, nor does the definition upthread.
This isn't a silly gotcha, these things matter in math. For instance, when solving a quadratic equation, allowing for the solutions to be equal lets you avoid special-casing your understanding - instead of memorizing when the equation has zero, one or two solutions, you just learn it has two or zero (real) solutions, and the two solutions are allowed to be equal.
It's perfectly reasonable understanding. Inequality isn't a natural implicit assumption. E.g. if I say you have two variables:
int a;
int b;
I doubt you'd be insisting that `a != b` at all times.
No. In the set of integers, 1 is not two divisors if repeated. It is one divisor.
An element to set relationship is that element A is or is not in set B. So, if the set of divisors only contains 1, there is only one divisor. If 1 and 1 made two divisors, 1 and 1 and 1 and… would make infinite divisors, rendering the concept of counting divisors (i.e., the cardinal number of the set of divisors) meaningless.
M is a divisor of N if it is a number that divides N without a remainder. While divisors can be negative, they are conventionally limited to non-negative integers in primality and factoring.
If you’d wanted to dig in on negative vs. positive divisors, that quickly provides an avenue for clearer formality, but piling on 1 and saying it’s not a silly gotcha is pretty fruitless. And please don’t bother to say “you didn’t say it has only two divisors”, as that would, again, be a silly argument.
So wind back and really formalize the definition if you want: A prime number is a natural number with only two divisors in the set of natural numbers, 1 and itself.
While set theory is axiomatic, it’s not practical for me (or anyone else) to explain conventional foundations to avoid someone feeling like they need to wiggle out of a prior bad argument.
Just say “ah, okay” or stop replying and move on. Feel free to read up in Wikipedia or any other texts (or ping me privately if you’d like to discuss further), but this thread isn’t looking like it’s going to meaningfully contribute to the broader discussion. Accordingly, I’ll leave it here unless something meaningful comes up.
