Is 2 Really the Oddest Prime Number?

[1MileCrash]

2 is the "oddest prime of all."

Regarding the old humorous "math joke" that 2 is the only even prime, thus it is the "oddest" prime of all. I have a bone to pick with this.

I don't think the idea of "even" numbers is any more special than numbers that are divisible by 3 or 5, or anything else. Divisibility by 2 just has a special name.

So we say that 2 is an 'odd' prime because it is the only prime that is even.
But, if we equivalently say that 2 is the only prime that is divisible by 2, we find that there is nothing special about this at all, because any prime p has the property that it is the only prime divisible by p by definition.

Thus, there is nothing odd about the prime number 2.

End rant.

 

[Tosh5457]

Well but even numbers are more important to group theory than numbers divisible by 3, 5, etc...

 

[Mark44]

One of the most important characteristics of integers is whether they are odd or even. It's arguably much less important that a given integer has a remainder of 1 when divided by 3, or a remainder of 0 when divided by 6, for just two examples.

 

[1MileCrash]

One of the most important characteristics of integers is whether they are odd or even. It's arguably much less important that a given integer has a remainder of 1 when divided by 3, or a remainder of 0 when divided by 6, for just two examples.

Why?

 

[lostcauses10x]

Thus, there is nothing odd about the prime number 2.

It as a single prime, takes out the largest amount of numbers from being prime.
50% of them greater than 2.

No other number unto itself can do that.

 

[1MileCrash]

It as a single prime, takes out the largest amount of numbers from being prime.
50% of them greater than 2.

No other number unto itself can do that.

I don't know what you mean.

 

[jbrussell93]

I don't know what you mean.

I think he means that since 2 is the only even prime, all other even numbers (50% of ALL numbers greater than 2) are unable to be prime. That is rather unique.

 

[AlephZero]

Many general theorems about primes are not true for 2, and often the reason is that the proof for a prime p involves the number p-1. There is no question that 1 is a "special" number (because. 1x = x, for all values of x), and the general proof may break down when p-1 = 1, or it may need the fact that p-1 is even, or composite, for all primes except 2.

Actually the same thing is true for theorems about fields in general (fields being a class of mathematical objects which include real and complex numbers as two specific examples), which apply to all fields except the finite field with only two elements. That's why field theory has many theorems that start "If ... and $1+1 \ne 0$ then ..."

 

[arildno]

Well, 2 is the first Prime Twin (version Siamese) to appear.
That's a fairly unique property!