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This question is often asked as "why is 2 a prime number" and the only answers I can find are "The definition of prime numbers is written in such a way that 2 is prime". Sometimes, if the question was "Why is 2 a prime number even though it's even" the answer will include some explanation that being even is not such a special property, it just means a number is divisible by two and of course two is divisible by two, that's the number itself and three is divisible by three and it's still prime and so on.

But there is something special about a number being even, and it 2 does have a special property that is not shared by any other prime number. It can be divided into equal subsets.

The answer I'm looking for will answer the question in the title and also answer: If we change the definition of a prime number, what effects does that have on mathematics theories, proofs, etc. Are certain claims that were useful and important under the old definition no longer true?

Specifically, What's the impact of changing the definition of prime to this: A prime is any integer that cannot be divided into smaller equal groups. N.B. this also changes the primeness of 1 but not any other number.

Segfault
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    It would mess up the theory of prime factorisation if $2$ weren't prime.... – Angina Seng Jan 25 '19 at 17:22
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    Why is $3$ a prime number? It's the only prime divisible by $3$. Shouldn't that be special? – saulspatz Jan 25 '19 at 17:22
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    A number $n$ is prime when has exactly two divisors : $1$ and $n$ itself. See Euclid's Elements, BK. VII, Def.11 : "A prime number is that which is measured by a unit alone." – Mauro ALLEGRANZA Jan 25 '19 at 17:23
  • @saulspatz I specifically addressed that in the question... – Segfault Jan 25 '19 at 17:25
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    There is nothing special about the ability to be divided into smaller equal parts. Every natural number $n$ can be divided into $n$ groups of $1$. – Lee Mosher Jan 25 '19 at 17:25
  • @MauroALLEGRANZA That's exactly what everyone says. My question is why – Segfault Jan 25 '19 at 17:25
  • The answer to "why" is that true theorems of number theory would become false, such as the theorem that every natural number has a unique factorization into a product of primes. – Lee Mosher Jan 25 '19 at 17:26
  • @LordSharktheUnknown yes, i suspect so. This question is asking how in specific detal – Segfault Jan 25 '19 at 17:26
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    There are numbers that have the property that they can only be divided by themselves and one. The number $2$ has this property so it is included in the set of numbers that have the property. Why would we discriminate against it? – John Douma Jan 25 '19 at 17:28
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    @Lee Mosher: Your comment showed up just as I was going to write the same thing, so I'll address another issue. "Are certain claims that were useful and important under the old definition no longer true?" --- All that happens is that certain results have to be expressed differently, such as saying "every prime number, along with 2, has the property that ...". It's like redefining a meter to be 1000 centimeters. Actual lengths won't change, but the numbers used to describe the lengths will change. For primes, the actual results known to be true will not change, just how they are described. – Dave L. Renfro Jan 25 '19 at 17:31
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    I will reiterate saulspatz point, but perhaps instead of $3$ I'll use $5$ as an example. You seem to think that $2$ is "special" because it is even. A two-even number is one which is a multiple of $2$. A two-even number has the special property that it always ends in one of $0,2,4,6,8$ (in base ten) and so is easy to spot. Well... $5$ is just as special. A number is "five-even" if a number is a multiple of $5$. A "five-even" number has the special property that it always ends in one of $0,5$ (in base ten) and so is always easy to spot. "Two-Even" and "Five-even" are just as special. – JMoravitz Jan 25 '19 at 17:31
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    Although, yes, $2$ has the additional special property that the relative density of the even natural numbers to the natural numbers is $\frac{1}{2}$, I don't find that at all surprising. You also have that the relative density of multiples "five-even" numbers to the natural numbers is $\frac{1}{5}$. Indeed, the relative density of "$p$-even" numbers is $\frac{1}{p}$. This doesn't actually have anything to do with prime-ness however, and so is a moot point. So... again, what is so special about $2$ to you? – JMoravitz Jan 25 '19 at 17:34
  • @JMoravitz I said that in the question... – Segfault Jan 25 '19 at 17:35
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    Why does everyone hate this question? It's a genuine question and I'm just trying to understand. – Segfault Jan 25 '19 at 17:36
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    It can be divided into equal subsets. --- I realize that you might have meant "divided" as split into half (like a sword slicing through the set only once, producing equal size parts), but in this case you're simply talking about integers divisible by $2$ and thus defining what you want to exclude in terms of itself (i.e. the argument carries little to no persuasive or rhetorical weight). I would have included this in my other comment, but ran afoul of the comment character limitations. By the way, I don't think the question is all that bad if you're just beginning to learn about primes. – Dave L. Renfro Jan 25 '19 at 17:39
  • Re JMoravitz comment of $\frac{1}{p}$ density: With regard to $2$-evenness there is only one type of oddness because division by $2$ gives two classes of numbers: those that have remainder $0$ (even) and those that have remainder $1$ (odd). Division by $p$ yields $p$ classes: one that is $p$-even and $p-1$ distinct classes of non-$p$-even numbers, according to their remainders which range from $1$ to $p-1$. For $p=5$, we get $5$-even numbers, but we also get $5_1$-odd numbers, $5_2$-odd numbers, $5_3$-odd numbers, and $5_4$-odd numbers. Each class is of equal density. – Keith Backman Jan 25 '19 at 19:21
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    Why do you say that $2$ is the only number that can be divided into equal subsets when $3$ can be divided into equal subsets? $3=1+1+1$, three equal subsets. – bof Jan 26 '19 at 03:06

6 Answers6

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We define things in such a way that they have some practical use, and help us make sense of the world around us. This is exactly how words and concepts are created, and also evolve. For example, we decided to give a name to a class of objects in the sky with similar behavior: 'planets'. These objects form what a philosopher might call a 'natural' class of objects. Having labels for them allows us to talk and think about those objects more easily, helping us with explanations, predictions, doing science, and again making sense of things in general.

But like I said, definitions can evolve: Pluto is no longer considered a 'planet', because after finding out more about our solar system we realized Pluto is in significant ways different from Neptune, Jupiter, Earth, etc. That is, by putting Pluto into different (though still related) class of 'dwarf-planets', we now look at it a little differently.

The same holds for mathematical definitions. For example, we could define 'huppelflup numbers' to be exactly those numbers that can be divided by 17 or by 631 ... but there just isn't much practical use to such a definition, and so we don't.

But the way we define prime numbers has lots of practical uses. For example, with the current definition, we get the nice, clean, result that every number has a unique prime factorization. And it's not just applications within mathematics that matter: prime numbers have tremendous importance for real life as well.

Now, if we were to exclude $2$ as a prime, this would no longer be true. And a bunch of other results would likewise have to be stated in a much more cumbersome way.

And by the way, this unique prime factorization theorem is exactly why mathematicians did exclude $1$ as a prime .. even though originally it was.

So yes, you're right that it is not as if definitions are fixed until the end of time. And maybe at some point in the future we redefine the sets of primes again to also exclude $2$, because doing so will have some other advantages.

However, I wouldn't hold my breadth: the current definition is very nice.

Bram28
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  • Thank you, that's exactly what I was wondering – Segfault Jan 25 '19 at 17:43
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    @Segfault I figured ... It's good you ask these kinds of questions, and hold our feet to the fire. For someone to say: 'but that's just what the definition is!" is really not enough; there are reasons to be made for defining things a certain way in the first place, and sometimes we may want to revisit those reasons and see if they are still most useful. – Bram28 Jan 25 '19 at 17:56
  • @Segfault I really want to stress the uniqueness property. If you can show that there must be a unique X for every Y, then you show that there is no arbitrary human choice involved. This often means that your X is "as good as it can get" in describing Y. You'll find this a lot. E.g. the invertibility of a matrix A says that for each y there is a unique x s.t. $A x = y$. The fundamental theorem of calculus states that there is a unique antiderivative up to an additive constant, etc. – WorldSEnder Jan 25 '19 at 21:35
  • My stochastic geometry Prof would always tell us Definitions, they are the rockstars of mathematics, not theorems – niko Jan 26 '19 at 01:40
  • "Planets" has changed far more than just Pluto and some similar older stories. It originally meant "wanderers" and referred to all objects whose position in the sky moved with respect to the fixed stars. The sun and moon were both planets at one time. – Paul Sinclair Jan 26 '19 at 03:17
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$2$ is considered a prime because most of the time that turns out to be convenient. $1$ is not considered a prime for the same reason.

That being said, $2$ is definitely the oddest prime.

Arthur
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You may wish to read the 2012 Journal of Integer Sequences article What is the Smallest Prime?, by Chris K. Caldwell and Yeng Xiong. The abstract starts with

What is the first prime? It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers.

Also, as already basically stated in the comments and other answers, the Introduction says

... whether or not a number (especially unity) is a prime is a matter of definition, so a matter of choice, context and tradition, not a matter of proof. Yet definitions are not made at random; these choices are bound by our usage of mathematics and, especially in this case, by our notation.

I enjoyed reading this article & found it educational. The only thing I have to add to this article & what's already been stated here is that I also believe it's generally good to question things instead of just accepting the status quo because "that's the way it is". During my years tutoring math at university, I had a philosophy of "there's no such thing as a stupid question, only a stupid answer". What I mean is that if the person has made a reasonable effort to resolve it on their own and the question is sincere, it's deserving of a reasonable answer.

John Omielan
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An addendum to the other answers:

The number $2$ is actually so special that it often is exluded from consideration. The set of primes with $2$ excluded is often referred to as the set of odd primes.

Imprecisely: We are "lucky" enough that we can describe this set using other accepted terms and expressions, otherwise we might have come up with a new name for this set.

Examples:

"Law of quadratic reciprocity — Let p and q be distinct odd prime numbers" - from Wikipedia page on Quadratic Reciprocity

"If a, b, c is a non-trivial solution to $x^p + y^p = z^p , p$ odd prime, then $y^2 = x(x − ap)(x + bp)$ (Frey curve) will be an elliptic curve" - from Wikipedia page on Fermat's last theorem.

"Let the characteristic of K be different from 2." - from Wikipedia page on Quadratic forms.

Gunnar Sveinsson
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Concepts which use prime numbers as objects for study and consideration are learned at a much earlier stage, in acquiring mathematical maturity, than sets. I don't think you mean "group" in the way that word is used in mathematics, but even in the way you do use, it's less useful than the indivisibility definition.

Plainly, there are many primitive concepts for which the decomposition of numbers into products of indivisible factors is telling something about their properties.

And there is the principle that definitions of terms are considered good if the theorems which can be phrased and proven using those terms are as simply-stated as possible.

grovkin
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Can you clarify what definition of the prime number you are using? The one I know is

a natural number that is divisible only by itself and 1

Or a variation of it. Number 2 expressly satisfies the rule and does not require any "special" ruling to be included.

Draco-S
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