3 ms·

Because the article doesn't actually say so (presumably because the author doesn't know the difference between "if" and "if and only if") the statement: (3n^3

by noqc 1y ago

Because the article doesn't actually say so (presumably because the author doesn't know the difference between "if" and "if and only if") the statement: (3n^3 − 13n^2 + 18n − 8)M_1(n) + (12n^2 − 120n + 212)M_2(n) − 960M_3(n) = 0 is equivalent to the statement that n is prime. The result is that there are infinitely many such characterizing equations.

deleted 1y ago
[deleted]
ijustlovemath 1y ago
A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.
- noqc 1y ago
  You could not be more wrong, and I am in a very bad mood unrelatedly, so that is all I will say.
  - ijustlovemath 1y ago
    feel free to expound once you're feeling regulated.
    - noqc 1y ago
      A iff B exactly means "A is the only way for B to be true". The rest of your comment is gibberish.
- lupire 1y ago
  I have no idea what you are trying to say. What is "this" and what is the other "this"? A and B? What is an arbitrary A?
  - ijustlovemath 1y ago
    Hm, not sure how much math you've had, so let's start with fairly basic stuff. A and B are statements within a given set of axioms whose truth value is knowable within those axioms. A could be something like "Some number k that we pick is prime", and B could be something like "k is even" When you see in math people saying "some statement is true iff some other statement is true", that "iff" stands for "if and only if", which really just means two things hold: 1. Starting with statement A, we can prove statement B. 2. Starting with statement B, we can prove statement A. In math shorthand, we'd write this as 1. A implies B, or just A => B 2. B => A You need to prove both directions for you to be able to say "A iff B" Let's try it with our example statements. Does it hold? Your intuition should be saying "absolutely not", and let's see why: 1. k prime implies k is either 2, or odd. So the statement A => B only holds when k is 2. We choose k, so this could be true in a trivial case, but does not hold in the generic case 2. k even implies k is divisible by 2, so again, the statement "k even => k prime" only holds for one trivial case and not in the generic one. Now for the original comment. I was pointing out that just because you have some proof of A iff B, does not mean there couldn't be another, completely separate statement C, for which you can prove A iff C. These relationships have equivalence, but are nonetheless not the same (outside of a categorical sense of sameness). Some of the most compelling math of the 20th century was showing the sameness of many different fields by finding new iff relationships.

by noqc 1y ago

deleted 1y ago

[deleted]

ijustlovemath 1y ago

A iff B doesn't mean "this is the only way for this to be true", it means A implies B and B implies A. B being the statement that a number is prime, but you can have any arbitrary A that is actually true.

noqc 1y ago
You could not be more wrong, and I am in a very bad mood unrelatedly, so that is all I will say.
- ijustlovemath 1y ago
  feel free to expound once you're feeling regulated.
  - noqc 1y ago
    A iff B exactly means "A is the only way for B to be true". The rest of your comment is gibberish.
lupire 1y ago
I have no idea what you are trying to say. What is "this" and what is the other "this"? A and B? What is an arbitrary A?
- ijustlovemath 1y ago
  Hm, not sure how much math you've had, so let's start with fairly basic stuff. A and B are statements within a given set of axioms whose truth value is knowable within those axioms. A could be something like "Some number k that we pick is prime", and B could be something like "k is even" When you see in math people saying "some statement is true iff some other statement is true", that "iff" stands for "if and only if", which really just means two things hold: 1. Starting with statement A, we can prove statement B. 2. Starting with statement B, we can prove statement A. In math shorthand, we'd write this as 1. A implies B, or just A => B 2. B => A You need to prove both directions for you to be able to say "A iff B" Let's try it with our example statements. Does it hold? Your intuition should be saying "absolutely not", and let's see why: 1. k prime implies k is either 2, or odd. So the statement A => B only holds when k is 2. We choose k, so this could be true in a trivial case, but does not hold in the generic case 2. k even implies k is divisible by 2, so again, the statement "k even => k prime" only holds for one trivial case and not in the generic one. Now for the original comment. I was pointing out that just because you have some proof of A iff B, does not mean there couldn't be another, completely separate statement C, for which you can prove A iff C. These relationships have equivalence, but are nonetheless not the same (outside of a categorical sense of sameness). Some of the most compelling math of the 20th century was showing the sameness of many different fields by finding new iff relationships.