Even vs odd numbers are not as important as we think they are. We could do the same to any other prime number. 2 is the only even prime (meaning it is divisible by 2) 3 is the only number divisible by 3. 5 is the only prime divisible by 5. When you think about the definition of prime numbers, this is a trivial conclusion.
Tldr: be mindful of your conventions.
Yes, but not really.
With 2, the natural numbers divide into equal halves. One of which we call odd and the other even. And we use this property a lot in math.
If you do it with 3, then one group is going to be a third and the other two thirds (ignore that both sets are infinite, you may assume a continuous finite subset of the natural numbers for this argument).
And this imbalance only gets worse with bigger primes.
So yes, 2 is special. It is the first and smallest prime and it is the number that primarily underlies concepts such as balance, symmetry, duplication and equality.
But why would you divide the numbers to two sets? It is reasonable for when considering 2, but if you really want to generalize, for 3 you’d need to divide the numbers to three sets. One that divide by 3, one that has remainder of 1 and one that has remainder of 2. This way you have 3 symmetric sets of numbers and you can give them special names and find their special properties and assign importance to them. This can also be done for 5 with 5 symmetric sets, 7, 11, and any other prime number.
Not sure about how relevant this in reality, but when it comes to alternating series, this might be relevant. For example the Fourier series expansion of cosine and other trig function?
Then you have one set that contains multiples of 3 and two sets that do not, so it is not symmetric.
You’d have one set that are multiples of 3, one set that are multiples of 3 plus 1, and one stat that are multiples of 3 minus 1 (or plus 2)
Oh yeah? What about 0? And 1?
Put them in a sieve of Eratosthenes and see what happens.
Spoiler, they aren’t.
They’re not prime. By definition primes have two prime factors. 1 and the number itself. 1 is divisible only by 1. 0 has no prime factors.
0 has all the factors. Itself and any other number.
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 1*1==1*1*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.
There is multiple things wrong here.
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1 is not a prime number because it is a unit and hence by definition excluded from being a prime.
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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.
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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)).
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(R,×) is not a ring (at least not in a way I am aware of) and not even a group (unless you exclude 0).
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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.
That’s why I wrote prime number instead of prime element to not add more confusion. I know that in general prime and irreducible are not equivalent.
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Oof, I remember why I didn’t study math 😅
But thanks for the explanationYeah, higher math is a total brainfuck :D You’re welcome.
I was never able to wrap my head around quaternions.
The meme works better if it’s 1 instead of 2. 1 is mostly not considered a prime number because a bunch of theorems like the fundamental theorem of arithmetic would have to be reworked to say “prime numbers greater than 1.” However, just because 1 is not a prime number doesn’t mean it’s a composite number, so 1 is a number that is neither prime nor composite.
2 is a prime number, but shit ton of theorems only apply to odd prime numbers, and a lot of other theorems treat 2 as a special separate case, because it behaves weirdly.
2 may be the only even prime - that is it’s the only prime divisible by 2 - but 3 is the only prime divisible by 3 and 5 is the only prime divisible by 5, so I fail to see how this is unique.
Exactly, “even” litterally means divisible by 2. We could easily come up with a term for divisible by 3 or 5. Maybe there even is one. So yeah 2 is nothing special.
“Threven” has a nice ring to it now that I think of it.
Gtfo! 😅
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I don’t get it, why does adding a hand move to the next prime?
It’s just the way the power rangers combined their forces
🚨 NERD ALERT🚨
Go define a vector space, nerd.
Go compute the p value of you being cool
Go integrate f(x)= 1/x on the domain (-1,1)
This is meme-ville population: me
Take a hike.
what why i’m serious i don’t get why the hands decrement the numbers
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.
Was trying for absurd. Didn’t mean to offend
what I don’t get is what the meme format’s supposed to mean, I can’t even find the name of it online
ohhhh so they’re power rangers, thanks
Spoiler: p < 0.05
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”
I kinda wish it was calculated for the world instead of the US though
2 is a prime though isn’t it
It is but if feels wrong
Yes, but it’s the only even one. Making him the odd man out
It pretends to be prime and we all go along with it to avoid hurting it’s feeling.
2 is a prime number though……
Is it Just because it’s the only even one?
And how is “even” special? Two is the only prime that’s divisible by two but three is also the only prime divisible by three.
Well 2 is the outlier because it’s the only even prime. It might not be “special” but it is unique out of all of the prime numbers.
“even” just means divisible by two. So it’s not unique at all. Two is the only prime that’s
evendivisible 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.It’s unique because all the others are odd numbers. This is crazy that you’re trying to argue this.
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”.
Often things hold true for all primes except 2. You come across things like “for all non two primes”
Like what? Genuine question, have never heard of this.
In the drawer in the living room in the house in my town in my state in my country.
Any examples? Sounds like you mean the reason why one is excluded from the primes because of the fundamental theorem of arithmetic.
I just remember it from numberphile, I don’t remember what videos sorry.
Wow that was fast I just edited my previous comment and you probably mean “1 and prime numbers” by numberphile with james grime.
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.
https://en.wikipedia.org/wiki/Fermat's_theorem_on_sums_of_two_squares
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.
Yo what about my man 9
Eyyyyy
7 ate them
9 isnt prime, it’s divisible by 7
just not very well…
Two is the oddest prime of them all.
2 is the prime supreme
@WolfhoundRO @HiddenLayer5, and 73 the one of Sheldon Cooper.
