# Does “evenness” have a deeper mathematical significance than being divisible by 2?

I have read and heard many times that “2 is the only even prime number”.

If I was to say “2 is the only prime number divisible by 2” it would be mere tautology; it follows inevitably from the definition of prime numbers, in the same way that 3 is the only prime number divisible by 3, 5 is the only prime number divisible by 5, etc.

Does being even have some deeper mathematical significance than being divisible by 2, or is “2 is the only even prime number” as tautological as “2 is the only prime number divisible by 2”?

• Yes and No but mostly no. Other than psychology "2 is the only even prime number" is no more significant than "17 is the only prime number divisible by 17". BUT 2 is the smallest natural number with more than unitary value may be seen as significant. It's very useful to consider a binary view of the world "things either are or aren't". Example: binary searching, power sets having $2^n$ elements, real numbers positive and negative duality, and simple ON/OFF toggling. But just how "deep" those are is very subjective. – fleablood Aug 26 '16 at 18:10
• For what it's worth, $-2$ is also divisible by $2$. – Mr. Brooks Aug 26 '16 at 21:20
• We calculate in base 10 which is factorized by 2 and 5. This means that everyone with no understanding of modulo or residue, will be able to see the pattern, because it appears that 2,4,6,8,0 will never be the last digit of a prime bigger than 2, and 0,5 never be last digits of primes after 5. This means you can use them as easy examples for explanations and everyone will understand them. It also means that nearly everyone will see the pattern. So the special thing about that statement is that it's the first in a series of tautologies. Which makes it the first that people spot. – HopefullyHelpful Aug 27 '16 at 7:20
• “All primes are odd, except 2, which is the oddest of them all.” — Graham, Knuth, Patashnik, Concrete Mathematics. – PM 2Ring Aug 27 '16 at 11:17

## 4 Answers

In practice, 2 does play a different role than other primes. There are many results (in number theory and algebraic geometry, for example) that are true in positive characteristic different from 2, but fail or are more complicated in characteristic 2. This is because a coefficient of 2 sometimes appears in some formulas (much more often than any other number, except 0 and 1), and in characteristic 2 that becomes zero, but in all other characteristics it isn't zero.

With that said, you will find for example some classification results (projective surfaces for example) that have a general answer, plus one or two more classes that exist only in characteristics 2 and 3, plus a couple more just in characteristic 2. (And, perhaps, also one or two ONLY in characteristic 3.) So 2 is not "really, really exceptional" in that way; 3 is also "exceptional" but less so than 2, and sometimes even 5 is a special case...

• Two is the smallest multiple natural number. Thus it "di"vides everything into a "di"chotomous "bi"nary state of one thing or another. This "di"chotomy of included/not included; considered/excluded; being/not being is pretty fundamental. But then again it is also pretty banal. – fleablood Aug 26 '16 at 18:33
• @fleablood - that is true, but off-topic; the OP asked about mathematical significance, not significance in general. – mathguy Aug 26 '16 at 18:50
• Mmmm... well, it's debatable. Take $|P(A)| = 2^{|A|}$ and not $b^{|A|}$. Why 2? Because it's a binary division; for every element there is a subset the element is in AND a subset the element isn't in. Why does a binary EITHER/OR have 2 options; why not b options? Because 2 is the smallest multiple natural number. I think that could be claimed to be mathematically significant. Maybe.... – fleablood Aug 26 '16 at 20:18
• This answer reminds me of this xkcd comic. – Will R Aug 26 '16 at 23:23
• Haha, does the chance of $2$ appearing in math follow Benford's law? Because it surely appears more than $3$ but less than $1$. – Simply Beautiful Art Aug 26 '16 at 23:29

It is exactly that tautological. By definition, a number is even iff it is divisible by $2$; that's all there is to it.

Very relevant is the Strong law of small numbers:

There aren't enough small numbers to meet the many demands made of them.

In the case of $2$:

There are two few even primes.

One could even say that $2$ is prime simply 'because' it is two small to have a non-trivial factor, being just adjacent to one. Similarly $3$ is prime simply 'because' it is right after $2$. Furthermore, every prime is the only prime divisible by it, simply by definition of prime. So $2$ is not special; $3$ likewise is the only prime divisible by $3$.

Also, see the excellent big list of Examples of Apparent Patterns that Eventually Fail and also a specialized question about the specialness of characteristic $2$.

"Two is the only even prime number" is only as meaningful as "2 is the only prime number divisible by two".

But I think there is a deepity to binary states. If we divide things into "things either are or are not" the pervades mathematics everywhere. $|P(A)| = 2^{|A|}$ because for each subset of $A$ and each element $a\in A$ either $a$ is in the subset or it is not. A sequence of "+" and "-" can be represent by $(-1)^{n}$ where the states are determined by whether $n$ is even or odd. Etc.

That's sort of deep. Kind of maybe.

But I think the deepidacity of it is not that "so and so is even" so much as "2 is the smallest non-unitary natural number". ... Actually, if I think about it, "2 is the only even prime number" is of absolutely no consequence but "2 is the smallest prime number" is.

So, I'll go out on a limb and say: Even numbers are "deep" because they are divisible by the smallest prime number and thus represent of state of dichotomy between "EITHER/OR".

But just how "deep" that is (or maybe it's banally inevetible --- ["but isn't the fact that it is banally inevetible it's self a deep statement about reality? WOOO! Trippy!"]) is highly subjective.

As a soft question it doesn't get much softer.

• I think you have both intentional and unintentional misspellings in here. – Robert Soupe Aug 27 '16 at 2:31

## protected by user99914 Aug 27 '16 at 20:37

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