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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.

closed as off-topic by Conifold, Swami Vishwananda, Mauro ALLEGRANZA, virmaior, Philip Klöcking Mar 29 '18 at 6:07

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  • 28
    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 3:34
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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 4:56
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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 7:09
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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 7:37
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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

10 Answers 10

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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.

  • 2
    I do not define vacuous truth as true. A vacuously true truth is vacuously true. – Achmed Mar 28 '18 at 4:14
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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 4:27
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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 4:57
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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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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.

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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.

  • 6
    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 9:03
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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 :-)

  • This is better than any of the other answers. I was just posting the same thing, but I see that you already did it. – Flounderer Mar 28 '18 at 15:32
  • 47 is an odd prime because it's the only prime divisible by 47. – user253751 Mar 28 '18 at 23:38
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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.

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    You are absolutely correct; I get the feeling that some philosophers dislike definitions :-( – Carl Witthoft Mar 28 '18 at 15:52
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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"

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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 9:16
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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 9:30
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    +1 Good geometric interpretation of what a prime means. – Frank Hubeny Mar 28 '18 at 11:15
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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
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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.

3

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).

2

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.

  • +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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