No, he’s right. “For any odd prime” is a not-unheard-of expression. It is usually to rule out 2 as a trivial case which may need to be handled separately.
It’s not unheard of no, but if you have to rule out two for some reason it’s because of some other arbitrary choice. In the first instance (haven’t yet looked at the second and third one) it has to do with the fact that a sum of “two” was chosen arbitrary. You can come up with other things that requires you to exclude primes up to five.
Okay? Like I said, it’s usually to rule out cases where 2 is a trivial edge case. It’s common enough that “for any odd prime / let p be an odd prime” is a normal expression. That’s all.
“even” just means divisible by two. So it’s not unique at all. Two is the only prime that’s even divisible by two and three is the only prime that’s divisible by three. You just think two is a special prime because there is a word for “divisible by two” but the prime two isn’t any more special or unique in any meaningful way than any other prime.
Of couse all the others are odd because otherwise they wouldn’t be prime. All primes after three are also not divisible by three… “magic”. The only difference is that there are is no word like “even” or “odd” for “divisible by three” or “not divisible by three”.
I’m picking on you because you’re looking for patterns where there are none. It’s a common meme format, and it just so happens that op wrote it like that.
let V be you mom’s vagina, a vector space over the field of pubes. We define my d as a vector such that d is in V. Thus my dick is in your mom’s vagina.
In this vector space p values are not defined, but I can assure you that my pp is > 9000.
The integral of f(x)=1/x from -1 to 1 does not converge, just like how your father is never coming back from buying milk. The principal value of that integral tho is 0, just like the amount of hugs you got as a kid.
Pretty sure that when we plug in a correction factor for the relative age of the Fediverse userbase, "today's lucky 10,000" becomes more like "today's lucky 10 million"
They are the cause of every pointless war in that region that we get involved in.
That’s like going out to the local pub and having that one buddy who constantly antagonizes the neighboring booth and then when the shit hits the fan, expects you to protect him, while he runs and hides under one of the tables.
Whenever I would buy rabbit food for a rabbit I was taking care of I would always get ads for chinchilla food and food for other small mammals. Like, I’m not out here collecting animals I just got the one.
Commonly primes are defined as natural numbers greater than 1 that have only trivial divisors. Your definition kinda works, but 1 can be infinitely many prime factors since every number has 1^n with n ∈ ℕ as a prime factor. And your definition is kinda misleading when generalising primes.
Isn’t 1^n just 1? As in not a new number. I’d argue that 11==11*1. They’re not some subtly different ones. I agree that the concept of primes only becomes useful for natural numbers >1.
How is my definition misleading?
It is no new number, though you can add infinitely many ones to the prime factorisation if you want to. In general we don’t append 1 to the prime factorisation because it is trivial.
In commutative Algebra, a unitary commutative ring can have multiple units (in the multiplicative group of the reals only 1 is a unit, x*1=x, in this ring you have several “ones”). There are elemrnts in these rings which we call prime, because their prime factorisation only contains trivial prime factors, but of course all units of said ring are prime factors. Hence it is a bit quirky to define ordinary primes they way you did, it is not about the amount of prime factors, it is about their properties.
Edit: also important to know: (ℝ,×), the multiplicative goup of the reals, is a commutative, unitary ring, which happens to have only one unit, so our ordinary primes are a special case of the general prime elements.
1 is not a prime number because it is a unit and hence by definition excluded from being a prime.
You probably don’t mean units but identity elements:
A unit is an element that has a multiplicative inverse
An identity element is an element 1 such that 1x =x1 = x for all x in your ring
There are more units in R than just 1, take for example -1(unless your ring has characteristic 2 in which case thi argument not always works; however for the case of real numbers this is not relevant). But there is always just one identity element, so there is at most one “1” in any ring. Indeed suppose you have two identities e,f. Then e = ef = f because e,f both are identities.
The property “their prime factorisaton only contains trivial prime factors” is a circular definition as this requires knowledge about “being prime”. A prime (in Z) is normally defined as an irreducible element, i.e. p is a prime number if p=ab implies that either a or b is a unit (which is exactly the property of only having the factors 1 and p itself (up to a unit)).
(R,×) is not a ring (at least not in a way I am aware of) and not even a group (unless you exclude 0).
What are those “general prime elements”? Do you mean prime elements in a ring (or irreducible elements?)? Or something completely different?
You’re mostly right, i misremembered some stuff. My phone keyboard or my client were not capable of adding a small + to the R. With general prime elements I meant prime elements in a ring. But regarding 3.: Not all reducible elements are prime nor vice versa.
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