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Since there are no integer numbers between two and one, how can two be divisible by a number other than itself and one?

Perhaps the definition of prime numbers is irrational and wrong, at least for two.

Why is 2 considered a prime number?

Additional Details:

I think a prime number is a number that is divisible only by itself and one when there is a number between itself and one, and this definition does not include the number two. A prime number must be able to be tested or tried by any number other than itself or one. All prime numbers are testable except 2. A prime number must be able to be tested by any number other than itself and one before being called a prime number. 2 cannot be tested by any number other than itself and one, therefore it cannot be called a prime number. The un-testability of the number of two makes it impossible to judge whether it is a prime number or not.

Andrew T.
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Achmed
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    Mathematicians say that because it is compatible with their definition, so two is a prime number, but I think it is undecidable. – Achmed Mar 28 '18 at 02:12
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    The answer to the title question is "because it satisfies the definition", i.e. it is divisible only by itself and one. There is no argument in the post as to why "prime number must be able to be tested by any number other than itself and one before being called a prime number" should be preferred instead, only dogmatic assertion. "Undecidable" has a different meaning that does not make sense here. So what exactly is being asked of us? – Conifold Mar 28 '18 at 03:34
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    "I think a prime number is a number that is divisible only by itself and one, When there is a number between itself and one" -- why do you make the arbitrary restriction to "when there is a number"? The first part (divisible only by itself and one) is enough to define the prime numbers, and is simpler. What's important is whether a number (other than 1 and self) that divides a candidate number exists or not -- not whether there are numbers to test to see if they divide. And in the case of 2, as with other primes, one doesn't exist. – Hejazzman Mar 28 '18 at 04:56
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    The def of prime number is (as you said) : "is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers." 2 is greater than 1 and it cannot be formed multiplying two smaller numbers. Thus, it is prime. Fullstop. – Mauro ALLEGRANZA Mar 28 '18 at 05:54
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    What you're saying is sort of akin to saying "I think airlines should only call it a direct flight when there is a closer airport they could actually stop at in between". But it's still quite reasonably a direct flight if you fly between two neighboring cities. People worried about direct flights only care if they have to get off... and applications worried about the primeness of a number only worry about whether some other number goes into it. Otherwise, for example, you couldn't use 2 in prime factorization! – JeopardyTempest Mar 28 '18 at 07:09
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    "A prime number must be able to be tested by any number other than itself and one" - why? Definition does not include that. – rus9384 Mar 28 '18 at 07:34
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    If 2 is not a prime, then that means that there are numbers that are not prime (all the powers of 2), yet do not have any prime factors, so goodbye Fundamental theorem of arithmetic, although perhaps you could argue that this theorem is not a "truth" but just something that arbitrarily depends on a particular definition of primality. – samgak Mar 28 '18 at 07:37
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    Perhaps the definition of prime numbers is irrational and wrong, at least for two. Well. Either millions of people are wrong, or you are wrong. What's more likely? – Eric Duminil Mar 28 '18 at 07:37
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    Can you please clarify your question? what makes you think that is 2 not testable? – Mehdi Mar 28 '18 at 08:17
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    @EricDuminil: Unquestioned pseudo-wisdom is dangerous. The mines, bomb-craters and unexploded bombs in my country - from two generations ago - know. Likewise, Newton, Einstein and Hawking knew (know?), too. Even if somethings sounds absurd because people tell me otherwise, one should verify. Or else: Remain cavemen. Anecdote: I make a second income from blogging, even though everyone said there's no money to be made in that. – phresnel Mar 28 '18 at 10:16
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    All numbers can be tested with any number provided infinity > n > 0 to see if they are prime. – Willtech Mar 28 '18 at 10:17
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    I get the reasoning here but for would agree for the reasons given that 2 must be considered prime. However,. for my analysis of the primes I treat 2 and 3 as not-prime but as axiomatic numbers. Then we can say that all primes occur at 6n+/-1, This is merely a convenience though, not an attempt to officially re-define the primes. –  Mar 28 '18 at 11:13
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    @phresnel: I really don't get what you mean, or what the connection should be to what happened in your country. There are many things I don't or cannot know in science, but I've been fascinated by prime numbers since I was a kid. I know and understand why 2 is prime, thank you very much. – Eric Duminil Mar 28 '18 at 11:18
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    @EricDuminil: You're right, I was too verbose. So tl;dr: Either millions of people are wrong, or you are wrong. What's more likely? is not a logical reason to not question something's validity. – phresnel Mar 28 '18 at 11:21
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    "I think a prime number is...". Well good for you, have fun with your own personal definition of what a "prime number" is. But to quote the Swedish humorist/wordsmith Tage Danielsson: "[N]aturally no-one is required to use all words in their agreed upon meaning, especially since neither you nor I had a say in that a loaf shall be called 'loaf', and cress 'cress'. You and I can for example agree to call a return ticket Stockholm-Malmö 'a tight evening dress with black tassels'. But I guarantee we will have hell to pay at the ticket slot at Central Station". – MichaelK Mar 28 '18 at 12:43
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    The main reason for having 2 as prime, but not 1 is because this yields an unique factorization on any positive integer into a product of (possibly multiple) prime numbers.

    In mathematics, edge cases are usually chosen such that the results are nice and elegant.

     – Discrete lizard Mar 28 '18 at 13:58
  • Better question: why would/should it not be? – user428517 Mar 28 '18 at 15:40
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    This question is definitely in the Humpty-Dumpty category. (see Through the Looking Glass; "a word means what I want it to mean") – Carl Witthoft Mar 28 '18 at 15:48
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    @PeterJ At least you understand enough to recognize that you've created a different algebra (different axioms). But since the standard algebra of numbers has a very clear definition of "prime," you are branching there. – Carl Witthoft Mar 28 '18 at 15:50
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    "Perhaps the definition...is wrong". Definitions can't be wrong, they're definitions. They are either useful or not. A definition of "prime" that excludes 2 would result in 1000s of math papers littered with phrases like "prime numbers and 2". Even if your definition made sense, there's a simple word for that too: "odd prime". – Lee Daniel Crocker Mar 28 '18 at 18:43
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    The OP's proposed definition is definitely not useful. It leads to the existence of numbers like 4, 6, 8, 10 etc. being composite numbers that can't be written as the product of prime numbers. – Monty Harder Mar 28 '18 at 21:15
  • Regarding testability: You can't just declare or assume that there are no integers between the number one and some other integer. You have to TEST to see if there are no integers between. If the answer is no, that is effectively a test for being prime. So the number two is testable for being a prime. – Syntax Junkie Mar 29 '18 at 02:29
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    It's weird to see this question on SE.Philosophy instead of SE.Math. Even ignoring the question of where it'd be most on-topic, it'd simply be more readily addressed on an SE that has TeX enabled. – Nat Mar 29 '18 at 04:56
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    @LeeDanielCrocker indeed. And 2 is the only even prime, which makes it pretty odd. – Eric Duminil Mar 29 '18 at 06:34
  • @RandallStewart is a test here necessary? I mean this "assumption" could qualify as an axiom? Per definition of N isn't 2 the number that comes after 1? – Mehdi Mar 29 '18 at 07:44
  • How long has 2 been considered a prime number? What I've been learning through the years working with algorithms and primes we come across an interesting anomaly with 2 being used as a prime when it really isn't because of its behaviors in mathematics. From what I understand there are algorithms that use 2 specifically approved by government agencies like RSA256/512 for example. What makes 2 interesting is because its an even number it creates predictable artifacts and anomalies in the encoded data that allows it to be decrypted without a key. Seems interesting this was pushed as the best..... – MichaelM Jul 09 '22 at 19:46

10 Answers10

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Mathematicians find the property

  • A non-invertible number that cannot be written as the product of two non-invertible numbers

to be more useful than the property

  • A non-invertible number that cannot be written as the product of two non-invertible numbers and is additionally larger than some other non-invertible number

That's why the former property has a name and the latter does not.

(when I say "non-invertible", I mean in the integers; 2 is not invertible, because 1/2 is not an integer)

In the grand scheme of things, this property is more properly called being "irreducible". Being "prime" is better described as:

  • A non-invertible number is prime if it has the property that, whenever it divides a product, it also divides one (or more) of the factors

It turns out that, for natural numbers, a number is prime if and only if it is irreducible.

In the grand scheme of things, invertible numbers are special things and need to be separated out to have a good theory of factorization. That's the real reason why the definition of "prime" excludes 1; because it is an invertible number.

Jo Wehler
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Why is 2 considered a prime number?

This is really a question of terminology. The current notion of an integer that is unrepresentable by a product of other integers is given the name "prime number," and you're asking why the term "prime number" doesn't refer to some other set of numbers.

Ultimately, it comes down to what is useful to mathematicians. Terminology isn't invented simply to pass the time; terminology is invented because somebody observes a concept repeated and then decides that that concept is common enough that it is worthwhile to give it a name. The current notion of prime numbers came up more often1 than the notion of "prime numbers excluding 2," which is why the former was given a short name and the latter made more ugly-looking.

The redefinition you're suggesting has occurred before—historically some mathematicians included 1 in lists of prime numbers, but it led to inconsistencies and required frequent usage of "prime numbers excluding 1," so eventually it became common not to include it. I like the quote given by MathWorld's article regarding this issue:

2 pays its way [as a prime] on balance; 1 doesn't.

How can two be divisible by a number other than itself and one?

You are right that it obviously can't, but it's fine to make arbitrary statements about the elements of the null set. You can say that every element of the null set is even and odd and counterexample to every unsolved problem in mathematics, because there is no element. Disallowing that kind of reasoning would be the same kind of inconsistency that happens when you exclude 2 from the prime numbers—you invent a special case that need not exist. Special cases being elegantly extrapolated from the common pattern is something to be celebrated, not eliminated.

1 Famous examples include Goldbach's Conjecture and the Fundamental Theorem of Arithmetic, both of which rely on the inclusion of 2 as a prime number to work.

Challenger5
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    This is the right answer. The definition of prime isn't arbitrary, it's one that is highly useful to mathematics. – Jack Aidley Mar 28 '18 at 09:03
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The other (perfectly good) answers reason from how prime numbers are usually defined in Mathematics. I will approach your question in a different way -- seeing what your definition leads to.

I think a prime number is a number that is divisible only by itself and one [exclusionary remarks omitted, for the sake of argument]. [A] prime number must be able to be tested or tried by any number other than itself or one. all prime numbers are testable [except] 2. A prime number must be able to be tested by any number other than itself and one before being called a prime number.

We might phrase the definition that can be deprehended from this passage, in an informal way, as: p is prime if there does not exist an n such that 1 < n < p and p can be divided by n. If p = 2, there are no (natural) numbers between 1 and p, and therefore there is no n satisfying the conditions above -- therefore, 2 is prime.

2 cannot be tested by any number other than itself and one, therefore it cannot be called a prime number، The untestability of the number of two makes it impossible to judge whether it is a prime number or not.

In this case, there being no candidates to test doesn't lead to undecidability, but to a vacuous truth.

P.S.: Note that my reformulation of your definition doesn't make 1 not a prime. See Challenger5's answer for some remarks on that issue.

duplode
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    I like your last statement

    In this case, there being no candidates to test doesn't lead to undecidability, but to a vacuous truth.

     – Achmed Mar 28 '18 at 04:04
  • Why did we define vacuous truth as true rather than false? – Achmed Mar 28 '18 at 04:12
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    I do not define vacuous truth as true. A vacuously true truth is vacuously true. – Achmed Mar 28 '18 at 04:14
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    One way of putting it: "All x that have the P property have the Q property" is equivalent to "There is no x that has the P property which doesn't have the Q property". If there is no x that has the P property, then the second proposition immediately follows. For some additional discussion, see Why did we define vacuous statements as true rather than false? and Vacuous Conditionals and How We Feel About Them, as well as the question I linked to in the answer. – duplode Mar 28 '18 at 04:27
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    @Achmed it's not up to us to "define" as true or false. A vacuous truth just means a self evident truth. – Hejazzman Mar 28 '18 at 04:57
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    @Achmed see https://math.stackexchange.com/q/2687297. Vacuous statements have the form (false statement) => (some other statement). This is true; we can deduce anything from a false statement. Maths doesn't put them in a separate class of vacuous truths as they can be proved like any other mathematical claim, even if it looks odd. E.g. in your question we want to prove (1<n<2) => (n does not divide 2). A proof: n is a positive integer and n<2, so n=1. So 1<1. Multiply by n to get n<n and hence n=/=n (n not equal to n). Let m divide 2, if m=n then n=/=n. So n=/=m for any m dividing 2, QED. – Simon L Rydin Myerson Mar 28 '18 at 08:27
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    I think you're really saying that it is a truism that 2 is prime. By definition something being a truism doesn't make it false! I think you could follow your line of thinking into a paradox which suggests that no number is prime. there are no numbers lower than 3 which can divide into it, so why call 3 prime, when it is impossible for it to be non-prime? – JeffUK Mar 28 '18 at 11:36
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    @JeffUK According to the view the OP had stated in the question, 3 being a prime is unproblematic, as there is at least one candidate divisor -- namely, 2. (This answer argues that that shouldn't be an issue.) – duplode Mar 28 '18 at 12:25
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    @Achmed Since words don't have some grand universal definitions beyond the ones we give them, you can define terms however you want to. But if your definitions don't agree with the ones everyone else is using, you might have some difficulty communicating with them. "Prime" and "Vacuously true" have the (common) definitions they do because those are the definitions mathematicians find useful. – Ray Mar 28 '18 at 13:15
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    @Achmed, you were told on Math.SE (by me) that the lack of candidates leads to a vacuous truth. VT was brought up elsewhere on the post but you never acknowledged it anywhere. Sigh. –  Mar 28 '18 at 17:59
  • Not all "vacuous truths" are "self-evident". "All integers strictly between 4 and 5 are equal to 7" and "All integers strictly between 4 and 5 are not equal to 7" are both considered "vacuous truths". –  Mar 28 '18 at 23:24
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2 is a prime because it makes Prime Factorization (a special case of integer factorization) possible and unique. By coincidence, this is the same reason why 1 is not considered a prime.

Suppose 2 would not be considered a prime. Then many theorems would have to include ugly modifications dealing with the case when a number has a factorization where a power of 2 appears. Mathematicians like to avoid that. When 2 is a prime everything just falls into place. Philosophically speaking, the current definition of primes is useful and elegant. When mathematicians want to exclude 2 for some argument, they simply use the term odd primes.

I share your sentiment that 2 is, in a way, an odd prime, because it is even :-)

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Jens
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Here is a definition: A natural number is called a prime number if it is greater than 1 and it cannot be written as a product of two natural numbers that are both smaller than it. https://en.wikipedia.org/wiki/Prime_number

Since 2 cannot be written as a product of two natural numbers that are both smaller than 2, 2 is prime by that definition.

Note that the definition explicitly excludes 1 although 1 has the very property, that is, 1 cannot be written as a product of two natural numbers that are both smaller than 1 because 1 is the smallest natural number.

One wants to identify the primes because any natural number can be factored uniquely into primes outside of order. That is why 2 is needed as a prime. It allows us to factor the even numbers.

However, if 1 were a prime the factorization would not be unique since 1 or any power of 1 would be a prime factor of the natural number. It is 1, not 2, that presents a potential problem. We have to deal with it explicitly and so exclude it from the set of primes.

Frank Hubeny
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Another definition of prime numbers is as follows:

A whole number greater than 1 which does not form a rectangle*

i.e. if you had two stones, you cannot arrange them into rows and columns to form a rectangle; therefore two is a prime number.

Unless you can find a way to make 2 stones form a rectangle, it is a prime number.

We can use this to address the 'philosophy' aspect of your question, you are effectively attacking the 'divisible by other numbers' aspect of your definition of prime, however my definition dispenses entirely with the concept of dividing by anything, and still gives us a complete list of 'prime numbers' which includes '2'.

Even in a theoretical universe where the number of numbers less than x is indeterminable (i.e. there may or may not be numbers less than x) it is still possible to independently verify whether or not x is prime using this method, therefore the 'primeness' of 2 can be verified regardless of the fact there are no whole numbers between 1 and 2

In fact, we can prove this is in our own mundane universe with an experiment:

  1. Have a friend sit on the other side of a curtain with a box of tokens.

  2. have them pick a number of tokens (x) but not tell you how many they have picked.

  3. Ask them if they can form a rectangle with those tokens.

  4. Repeat the experiment, ensuring that at some point they choose 2 tokens

You will, between you, confirm that 2 is a prime number.

*: : By my definition 2x1 and 1x2 are 1-dimensional lines, not rectangles, and maps to our other definition of 'only divisible by 1 and itself'. If you think these are rectangles, you could add "... rectangle with both sides > 1 to the definition"

JeffUK
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    What is really good about this definition is that you can understand it without having to understand division. – Oscar Bravo Mar 28 '18 at 09:16
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    @OscarBravo You also don't need to have a common number system and it proves that prime numbers are physical, not some abstract mathematical concept. I like the fact you can teach it to kids about the same time they learn to count with tokens! – JeffUK Mar 28 '18 at 09:30
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    +1 Good geometric interpretation of what a prime means. – Frank Hubeny Mar 28 '18 at 11:15
  • @OscarBravo that actually is more important point than I realised in answering OPs question. It breaks their assumption that you need to test primes by division which is a key premise of their argument – JeffUK Mar 28 '18 at 12:00
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    This fails because a rectangle of dimension N-by-One is legal in a lot of usages. The fact that MATLAB does this is the bane of my existence. – Carl Witthoft Mar 28 '18 at 15:53
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    @CarlWitthoft See the footnote; I wrote it especially for you in advance – JeffUK Mar 28 '18 at 17:03
  • @CarlWitthoft - if you consider the dimensions of the rectangle formed by the stones to be the distances between the centres of the farthest stone in each direction, then a 2x1 arrangement of stones has a size of 1x0, which is indeed a line and not a rectangle... – Jules Mar 28 '18 at 18:31
  • @Jules you're preaching to the choir. Problem is that some people are not following (e.g. MATLAB) – Carl Witthoft Mar 28 '18 at 18:31
  • @CarlWitthoft - ah... I haven't used MATLAB since I was an undergraduate, more decades ago than I care to admit. :) – Jules Mar 28 '18 at 18:32
  • @jules that's where I was coming from with my definition, but "Form a rectangle of 0-dimensional points" is a little clunky! – JeffUK Mar 29 '18 at 08:09
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From Wikipedia:

A prime number is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers

there is only one natural number smaller than 2 and it is 1

1*1 = 1 != 2

2 cannot be formed by multiplying two smaller natural numbers

2 is a prime number.

Mehdi
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A mathematically-precise way of expressing "Every number can be written uniquely as a product of primes" forces us to include 2 as a prime number.

Let Q be a set of natural numbers such that:

  1. Every natural number can be written as a product of elements of Q.
  2. If a natural number n can be written as the product over all q in Q of q to the power a(q), and n can also be written as the product over all q in Q of q to the power b(q), then a(q)=b(q) for all q.

Then Q must be the set of prime numbers. (This is easy to prove using the properties of prime numbers).

If 2 was not a prime number, then we wouldn't have this definition of the primes (because it would not be possible to write 2 as a product of primes). If 1 was a prime number, then we also wouldn't have this definition (because there would be more than one way to write every number as a product of primes).

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All numbers can be tested with any number provided p? / {∞ > n > 0} = prime? provided n is a natural number to see if they are prime. I know my hieroglyphics are not math but you know what I mean.

Note: Actually, I will accept an edit if someone can actually write that correctly.

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  • +1 You are making an important point. It is not just numbers less than some natural number, but the set of possible divisors of a natural number are any numbers greater than 1. This set is not empty even for 2. The only correction I would make is n should be greater than 1, not 0. The number 1 will always divide a natural number since it is a unit. – Frank Hubeny Mar 28 '18 at 13:02
  • @FrankHubeny I am hesitant to edit on that basis even though the result of diving by one is a foregone conclusion, it can still be tested just as numbers larger than a potential prime can be. I do see your point. – Willtech Mar 28 '18 at 18:29
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A natural number n > 1 is prime if it satisfies the following property:

For all natural numbers a and b in the interval [1,n], n = ab implies n = a or n = b. For the case n = 2, this involves quantification over the nonempty set [1,2]. Thus, it isn't true that the only way to define prime numbers is in such a way that it is only vacuously true for n = 2.

In fact, there are many ways to characterize primes, most of which involve nonvacuous quantification even in the case of 2. For example, since the converse of Euclid's Lemma is true, it could be taken to be an alternative definition of prime: p > 1 is prime if and only if for all integers a,b if p divided ab then it divides either a or b.

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