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0 pointsTeMPOraL2y ago0 comments

> Two divisors for positive integer X: 1, X

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.

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chaboud2y ago

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.

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