Oh yeah? What about 0? And 1?
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.
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?
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.
Yo what about my man 9
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.
Then you have one set that contains multiples of 3 and two sets that do not, so it is not symmetric.
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?
I don’t know if it’s intentional or not, but you’re describing cyclical groups
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 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.
Often things hold true for all primes except 2. You come across things like “for all non two primes”
Any examples? Sounds like you mean the reason why one is excluded from the primes because of the fundamental theorem of arithmetic.
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
