Why isn't 1 a prime number?

https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/

by gpvos • 7 years ago

273 125 7 years ago

blogs.scientificamerican.com

earthicus2• 7 years ago

Very nice article! Also of interest, 1 not being prime can be viewed as a particular example of a more general phenomenon that occurs throughout mathematics, known as the 'Too simple to be simple' principle:

https://ncatlab.org/nlab/show/too+simple+to+be+simple

bordercases• 7 years ago

Could you have discovered the example through use of the principle?

Or likewise, does the principle let you extend to other conclusions?

earthicus2• 7 years ago

Do you mean the example of 1 not being a prime, or the example in the article of Z[root -5] not having unique factorization? You could certainly discover the former, but not the latter as far as I can see. The idea is that the principle should guide you on what the 'right' set of definitions are, so that your theory detects interesting theorems and isn't riddled with edge cases. You might make an analogy to programming, finding good primitive data types and interfaces so that you can implement elegant and efficient algorithms and well structured modules.

Perhaps the key insight of the 'too simple to be simple' principle is that we want the definition to impose both existence & uniqueness of something. In this case proper divisors (divisors strictly less than the number itself e.g. the proper divisors of 6 are 1,2, & 3).

-------------------------------

Let me give a concrete example of how it might be used.

- Uniqueness is captured algebraically here by the notion of a 'subsingleton' set: a subset where any pair of elements are equal. This can happen if (1) the subset has only a single element, or (2) the subset has NO elements (in which case the uniqueness requirement is vacuously true).

- Existence is captured by the notion of a 'singleton' set: a subset with exactly 1 element: any two elements belonging to the set are unique, AND such an element exists.

First lets apply this to the definition of the regular primes: the set of proper divisors of 1 is the empty set. The set of proper divisors of any other prime is the singleton set {1}. Thus the naive definition says 'proper divisors of primes form a subsingleton': it asserts they are unique, but not that they exist. The more sophisticated definition of the primes (which excludes 1) asserts the proper divisors of primes form a singleton: uniqueness and existence.

Now lets try to apply it (somewhat informally) to the fictional example of even numbers and 'even primes'. Examples:

   2 is an even prime, 
   4 = 2*2 not an even prime
   6 is an even prime, 
   8 = 4*2 is not, 
   10 is even prime,
   30 is an even prime is as well, and so on.

Here I have uniqueness of proper divisors (they form a subsingleton), but only because existence of proper divisors fails for every even prime (not just 2)!

Now what happens if we try and factor 60?

   60 = 2*30
   60 = 6*10

It not great surprise that uniqueness of prime factorization fails as well (one of several problems with this informal example). The principle didn't help me find this example, it suggests that if I use the above notion of an 'even prime', i'm not going to get a good set of theorems.

gabrielblack• 7 years ago

Bad example, if 1 isn't prime, 2 x 1 isn't a product of prime, so not all the number can be written as product of primes. My point is that you can't write an introductory article, supposed to be understandable to the masses, using a form able to confuse further the readers. A student, reading this, I don't think will understand this explanation that seems contradictory.

"My mathematical training taught me that the good reason for 1 not being considered prime is the fundamental theorem of arithmetic, which states that every number can be written as a product of primes in exactly one way. If 1 were prime, we would lose that uniqueness. We could write 2 as 1×2, or 1×1×2, or 1594827×2. Excluding 1 from the primes smooths that out."

kdmccormick• 7 years ago

All positive integers can be written as a product of primes, where a product is a sequence of numbers that are multiplied together. In the case of a composite number like 45, the primes are (3,3,5).

When the sequence has one element, then that element is the product. So, 2 is the product of (2).

When the sequence is empty, the product is 1.

theoh• 7 years ago

This is explained more formally here: https://en.wikipedia.org/wiki/Empty_product

skoskie• 7 years ago

As a total layman, I think this is my favorite definition so far.

mintplant• 7 years ago

Furthermore, the ordered sequence of primes that multiply to any given positive integer is unique. If 1 were considered prime, that wouldn't hold, as you could keep adding as many 1s as you like to the start of the sequence: (1,3,3,5), (1,1,3,3,5), etc. Thus 1 encodes no useful information here.

zamadatix• 7 years ago

It's a good example just a poor rendition of the definition of the fundamental theorem of arithmetic. The proper definition includes "either prime or the product of primes". Many mathematical texts shorten this to "the product of one or more primes" which doesn't make much sense to me but if you take it as acceptable to use "product" with a single number then some shorten it even further to "the product of primes" (primes now representing "prime numbers" not "multiple primes") resulting in what you see in many places/this article.

Certhas• 7 years ago

It does make sense in a mathematical context though. You might write something like

p_S = \prod_{x \in S} x

And then later find out that the set S contains one element. Or it might even be the empty set in which case you'd interpret the above as equal to 1. This makes it work that

p_{S union S'} = p_S p_S'

in all cases.

amluto• 7 years ago

Surely “product of primes” means “product of one or more primes”.

globuous• 7 years ago

Yeah, but it also says "in a unique way" and 1 breaks that since it sends any other number to itself under multiplication. So maybe a prime number also has the property that it sends any other number not to itself when multiplied by ? maybe that's already in it ?

iamdead• 7 years ago

Product of zero or more primes. Otherwise you cannot factor 1.

detaro• 7 years ago

Sloppy wording, yes, bad example, not really. "just" requires you to consider "2" to be the product of all numbers in the multiset {2}. Let "product of primes" mean "product of one or more primes" and it works.

maire• 7 years ago

I thought the unique product of primes argument was great!

I assume most people on this list are engineers. They have college level math but not necessarily number theory. Number theory requires a similar thinking to the problems we solve in all the time in engineering fields.

I just explained the unique product of primes argument to my mechanical engineering husband. He got it right away and also thought it was awesome!

SkyBelow• 7 years ago

I think the best way it was explained to me was say that both exist as sets of numbers, which for now we will call Set_1 and Set_2.

Set_1 includes 1, Set_2 does not.

We have two possibilities for naming.

Option A: Set_1: Primes. Set_2: Primes except for 1.

Option B: Set_1: Primes and 1. Set_2: Primes.

You can then show the differences in the properties that make having Set_2 have the default name an easier to use convention than letting Set_1 have the default name. But at the core is the understanding that this is an arbitrary distinction done just for ease of use. Had we called Set_1 the Primes, or if ever someone became Galactic Emperor for a day and made the change, then Set_2 would get a new name for ease of use and we would just use it instead.

There are many other sets that are really close as well, which have their own properties.

Side note, SET_1 is listed on oeis https://oeis.org/A008578 and still includes 1. It is called "Prime numbers at the beginning of the 20th century".

bensalem• 7 years ago

Just the definition.

a number with no factors beside 1 and itself, excluding zero

1 does not meet that definition, since 1 is already counted, the other factor needs to be different.

mcphage• 7 years ago

Sure, but the next question—which the linked article is about—is, why is that the definition?

HocusLocus• 7 years ago

The number 1 is a prank by Loki.

frankbreetz6• 7 years ago

Funny true story about this: One time a man came up to me and handed me his phone, and asked me to call his mom. He was about to pass out and he asked me not call an ambulance (God bless America) , he appeared to have a concussion. By this time there was about 10-15 people around him and he could barely talk. I asked him what his phone code was and instead of just giving it to me he said "it's the first four prime numbers". Immediately, about five people shout "1,2,3,5". I am no longer holding the phone, because I handed to someone else to make sure it was okay. Sure enough, I was in a mathematical proofs class and we had just discussed this topic. So, I say "one is not a prime number". Of course, we get the phone unlocked in the second try with "2,3,5,7" and the guys mom is on the way. Everyone thought I was a genius a hero.

rytill• 7 years ago

That’s great. Thank you for sharing.

Seems a rather cryptic way to go about things with your health on the line.

kzrdude• 7 years ago

It reveals how he thought about his code, and he didn't have the presence to "compute" what it would be in digits.

frankbreetz6• 7 years ago

Yeah, he was basically laying on the ground and barely got the phrase out.

hackerman12345• 7 years ago

Reminds me of The Riddler...

wutbrodo• 7 years ago

Well he had a concussion..

judge2020• 7 years ago

Thankfully iOS added an "emergency" screen you can set up to show your relatives and their phone numbers without unlocking the phone.

sfg• 7 years ago

Nicely done.

https://xkcd.com/208/

speg• 7 years ago

You can also dial emergency contacts without unlocking the phone. They are accessible from the medical ID page on iOS, I assume Android has similar.

snazz• 7 years ago

Few people have set up that feature from what I’ve seen.

westurner• 7 years ago

You can also dial emergency contacts without unlocking the phone. They are accessible from the medical ID page on iOS, I assume Android has similar.

You can set a Lock Screen Message by searching for "Lock Screen Message" in the Android Settings.

You can also create an "ICE (In Case of Emergency)" contact.

gnicholas• 7 years ago

Or just tell Siri to “call my wife/mom/son” or “redial last number” to start figuring out who the person is.

I wonder if dialing 911 from their phone would give emergency services some helpful info. Do they automatically get caller ID (perhaps enhanced)?

jammygit2• 7 years ago

As far as mild-moderate concussions go, there is unfortunately not much to do by way of treatment besides low stimulation rest.

I hope this person was okay. Concussions are truly hell when they don't clear up nicely.

Side note, I was told once by an occupational therapist specializing in brain injury that most concussions are from standing up from a crouch in the kitchen and hitting your head on a corner of a cupboard (MVA/sports are very common too). I didn't believe her at the time, but I don't write it off either.

frankbreetz6• 7 years ago

We got ahold of his mom, and one of the bystanders was a nurse. She didn't seem too worried, so I think everything was okay. We were at a sports park and I think he hurt himself playing soccer.

philwelch• 7 years ago

My wife got a concussion almost exactly that way except it was the dryer door.

IgorPartola• 7 years ago

This is why I always close all cabinets and only open one at a time.

stcredzero• 7 years ago

most concussions are from standing up from a crouch in the kitchen and hitting your head on a corner of a cupboard

I ended up injuring my back and walking in the Early Asimo robot's "old man poopy pants gait" for an entire half a year, following a sneeze. You can google that. It's quite common to injure your back after a sneeze.

episteme• 7 years ago

This is the most "that happened" comment I have ever seen on HN.

frankbreetz6• 7 years ago

I don't know what to say... There is no way I am clever enough to make something like this up. Maybe when I embellished the hero and genius part.

AnIdiotOnTheNet• 7 years ago

It does stretch credulity quite a bit, but I've definitely seen a lot worse.

jvln• 7 years ago

will have to change my phone code.

billpg• 7 years ago

"one is not a prime number"

An odd royal and mathematical version of The Prisoner?

blu422• 7 years ago

"Whenever a product m×n is divisible by p, then m or n must be divisible by p."

Actually it's the other way around:

'Whenever m or n are divisible by p, then their product m×n must be divisible by p.'

The opposite is not true, i.e. the statement is not reversible and still true. For instance:

6x4 = 24; 24 is divisible by 8, i.e. 24 mod 8 = 0, yet 6 mod 8 = 6, and 4 mod 8 = 4.

[ed] I've stepped into a cognition discontinuity, move along, nothing to see.

jcranmer• 7 years ago

If p is a prime number, then the product m×n being divisible by p means that either m or n is divisible by p. In fact, this is how prime numbers are defined when generalized to generic integral domains.

blu422• 7 years ago

Doh, my reading balked at the embedded add. Yes, p being a prime actually inverts the statement successfully. My bad.

camjw• 7 years ago

8 is not prime

pvg3• 7 years ago

"M₆₇ is not a prime" was also a famous single-slide silent lightning talk.

https://en.wikipedia.org/wiki/Frank_Nelson_Cole

crazygringo• 7 years ago

tl;dr: "...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."

In other words, because semantics.

fjsolwmv2• 7 years ago

That shows the author doesn't understand the issue well, because every definition is semantics. That's what the word means. The question is why certain semantics are prefered.

Hackbraten• 7 years ago

And, even more important: is it Numberwang?

HenryBemis• 7 years ago

Wikipedia [1] also mentions that: "Most early Greeks did not even consider 1 to be a number, so they could not consider its primality". Both articles (scientific american & wikipedia) refer to the same paper [2].

[1] https://en.wikipedia.org/wiki/Prime_number

[2] https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h...

empath75• 7 years ago

I always raise an eyebrow at statements like ‘most early greeks’, because who knows how many of them actually read Euclid or had any exposure to math at all.

pcmaffey• 7 years ago

I've always thought of 1 as the definition of what a prime number is. Therefore, because it's generally bad practice to define a thing in terms of itself, 1 shouldn't be categorized as a prime number. Instead, it's like the Aristotilian Ideal of prime made linear.

dmarlow• 7 years ago

Interesting read. I'm an engineer and had considered 1 to be prime.

I guess you can say that 1 is just an absolute unit.

antaviana• 7 years ago

I understood prime as the minimal set of numbers with which you can generate all natural numbers by multiplying them.

There are infinite natural numbers and infinite prime numbers, but you need each and every prime number to generate any other natural number.

So I'm puzzled to learn that 1 is not a prime number because I do not know how to generate 1 if we exclude 1 from the prime set.

ummwhat• 7 years ago

For that matter, even with 1, how do you generate 0?

roywiggins• 7 years ago

Zero being a member of the natural numbers or not depends on convention.

OscarCunningham4• 7 years ago

There are actually many circumstances in which we do want to treat 0 as a prime. For example it generates a prime ideal in the integers.

One way to state unique prime factorisation including 0 is to treat the set of primes as a pointed set with 0 as the point. The smash product makes the category of pointed sets into a symmetric monoidal category. Then unique prime factorisation can be stated by saying that the multiplicative monoid of natural numbers (with 0) is isomorphic to the free commutative monoid object on the set of primes.

yakubin3• 7 years ago

Prime numbers are defined as elements of rings, so we need two operations, the other being addition (understood as a group operation). So you generate 0 by 1-1.

OscarCunningham4• 7 years ago

The number 1 is what you get if you multiply no numbers together. Just like if you add no numbers together you get 0.

davnicwil4• 7 years ago

Can't you divide any number by itself (multiply by its inverse) to get 1? Or does that not count?

codethief• 7 years ago

With the exception of the number 1, the inverse of any natural number with respect to multiplication is not a natural number.

tobiasSoftware• 7 years ago

The problem is that the usefulness of primes comes from the ability to break down a number into a unique set of primes, and including 1 as a prime defeats that usefulness. While your understanding of primes is an elegant idea, in sciences, usefulness trumps elegance.

From a programmer's point of view, a similar thing happens with Lisp. Lisp is a very elegant language due to its minimal structure, but not a very useful one with respect to how often it is used.

kemitche• 7 years ago

It almost seems to me that there are two very similar, overlapping mental models* of "primeness".

The first is: "numbers that can be multiplied together to get all other numbers". This mental model by necessity excludes 1.

The second is: "numbers that cannot be 'made' by multiplying other, smaller numbers together". This mental model is "looser," and ends up including 1, and maybe 0 and -1.

The first mental model tends to be more mathematically useful, so becomes the official definition of prime.

\* Note that I explicitly use "mental model" here instead of "definition," because I am discussing different ways that different humans try and _understand_ different sets of numbers.

Sniffnoy• 7 years ago

1 is the empty product. You generate it by multiplying nothing together.

(Or to put it differently, the set of primes forms a generating set for the monoid of natural numbers under multiplication, not the semigroup. Semigroups are kind of dumb. :P )

bo10242• 7 years ago

I was glad that the article got into the definitions of irreducible, prime, and unit in more general algebraic structures! This question is all about these three concepts.

A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1. So these are units in the integers. Units are really important, but they don't get you anywhere on their own. We can think of non-units as building blocks.

So then the question comes up which building blocks can be broken down into smaller blocks, like "times 6" can be divided into "times 2, then times 3". Using a unit doesn't count, because you're still equally far from the starting point (so 6 = (-6)(-1) is not really breaking down 6).

Blocks that can't be broken down any farther are irreducible (like 7). But in common parlance, that's what we think of as the definition of a prime number. It doesn't include units like 1, because they're already broken down.

More generally, being prime isn't about being broken down, it's about building up. Suppose I tell you that n is divisible by 2, and n can be broken down into two pieces, n = ab. Then you know that at least one of a or b is divisible by two. So two is prime because anything that is built from two, when broken down into pieces, has a piece that can be built from two. (Example: 6 divides 12, but you can break 12 up into (3)(4), so 6 isn't prime.) This definition of prime also doesn't include units like 1 because they're not building blocks in the first place. Everything's already divisible by units.

Prime integers have both the prime property and the irreducible property. Either way, I think it makes sense that 1 isn't included -- being prime and irreducible are properties of building blocks, and 1 is a unit.

red_trumpet• 7 years ago

A unit is an operation you can apply multiple times and get back to the starting point, for example multiplication by 1 or -1.

Nope. That is an idempotent[1], or a root of unity, i.e. x^n = 1 for some n >= 0. A unit is something which has an inverse, though that inverse need not be a power of x itself. For example, 1/2 is inverse to 2 in the rational numbers.

[1] https://en.wikipedia.org/wiki/Idempotence

bo10242• 7 years ago

Oops, thanks! (been a while) Those are examples of units, but units are more generally "building blocks of 1" (if you apply a unit, there is some other unit that gets you back to 1). Thanks again for the correction.

legohead• 7 years ago

When attempting to prove intelligence to aliens by mathematical proof, would we include 1 as a prime number?

mcguire• 7 years ago

Mathematicians: No!

Engineers: Eh, maybe?

The guy doing the engraving on the gold plaque on the side of the space probe: What is that, a potato?

Aliens: They treat root vegetables as numbers.

Alien anthropologist: They worship plants.

Alien TV presenter: In showing us agriculture, they introduced us to mathematics!

(Me: I think I need to cut down on the cold medicine.)

fjsolwmv2• 7 years ago

It doesn't matter. An additional element at the beginning of a sequence won't make anyone think you unintelligent, just a bit confused

jmmcd2• 7 years ago

Anyone who says 'zed' and indeed cheerio, pip pip, won't say "negative five". They'll say "minus five", of course.

ColinWright4• 7 years ago

That depends - this divide is not just "USA English Speaking People" versus the rest. Within the UK, Australia, New Zealand, and Canada, there is a split between people who say "minus 5" and people who say "negative 5".

For some "minus five" is an operation, not a number, and for others "negative five" is meaningless, the correct expression would have to be "the negative of five". There are entire PhD theses written on the topic of how to avoid confusing young children when introducing them to numbers less than 0, and the words we should, and should not, use in that context.

jmmcd2• 7 years ago

I was thinking more of UK (the only place where "pip pip" makes sense as a stereotype) versus the rest! Are you saying there are those in the UK who say "negative five"? That would surprise me.

gerdesj• 7 years ago

Yes, and who on earth says "cheerio, pip pip"? "Toodle pip" or "cheerio" or "pip pip" are acceptable.

Those PhD theses seem a little misguided (if well meaning) to me. Language is riddled with ambiguity - "panda eats shoots and leaves" for example. Despite that ambiguity we generally manage to get along. Teaching critical thinking in general might be a better idea than fixating on specific issues.

rdl• 7 years ago

There are also a bunch of cases where you specifically want "odd primes" (i.e. excluding 2).

Someone4• 7 years ago

The problem here is that “is not prime” doesn’t imply “is composite”. The number 1 is neither prime nor composite. Why? Because that is the choice that makes more math look nicer. If 1 were considered a prime, lots of theorems would have to say “let p be a prime >1”, rather than “let p be a prime”.

Mathematicians have struggled with this for centuries before agreement was reached. See “The History of the Primality of One: A Selection of Sources” (https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.p...)

The “math looks nicer” argument isn’t unique to this example. It also is, for example, the ‘reason’ that 0⁰ equals 1, or that there are as many integers as rationals.

pvg3• 7 years ago

In what way is the last one a matter of making it 'look nicer'? Either there is a bijection between these or there isn't - that seems like a concrete thing that's not a choice of terminology or aesthetics.

Someone4• 7 years ago

By making the choice that that applies to infinite sets, too, you go against strong intuition such as that, for example, there are twice as many integers as even numbers (that intuition is very strong; even math students will start out thinking it is true, not “I wouldn’t know”)

nimih2• 7 years ago

The “math looks nicer” argument [...] is, for example, the `reason' [...] there are as many integers as rationals.

I'm curious what you mean here: The integers and rationals have quite different algebraic and geometric properties, and the fact that they have the same cardinality has always struck me as a deep result, rather than just a matter of definitions.

pvg3• 7 years ago

Just occurred to me they were perhaps thinking of https://en.wikipedia.org/wiki/Continuum_hypothesis and mis-remembered/typed.

Someone4• 7 years ago

I meant that mathematicians struggled with infinities for quite a while. Chances are they initially modified

  “two sets have the same size iff there’s
   a 1:1 mapping between items in the set”

  “two *finite* sets have the same size iff there’s
   a 1:1 mapping between items in the set”

before ‘bending’ the meaning of ‘same size’ a bit to allow a strict superset of another set to be of equal size as that set.

mirimir• 7 years ago

I was going to say "because we said so", but this is a far^N more interesting comment. Elegance is a very useful criterion in many areas of mathematics, science and art. Stating the obvious, I know, but still.

Cogito• 7 years ago

The point of the article is that we don't consider a unit in a number system to be prime, and in the integers/rationals/reals 1 is a unit.

Why we don't consider units to be prime is more than just it "makes more math look nicer."

As discussed in the article, once we explored different types of number system (the example given is an adjoin of the square root of negative 5 to the integers) we were able to classify numbers that no-one would ever 'intuit' as prime: the units.

The defining characteristic of these non-primes is shared by '1' in the integers/rationals/reals so it is consistent to call 1 a non-prime as well.

strictnein• 7 years ago

It also is, for example, the ‘reason’ that 0⁰ equals 1,

Is it? That seems different. The math proof for that is pretty easy.

a^n * a^m = a^(n+m)

Is the argument just that we should say the above is true if n or m != 0?

OscarCunningham4• 7 years ago

Right, we want to define 0^0 in a way that fits in with the usual rules of exponentiation. The problem is that 0^n is always 0 for nonzero n, but a^0 is always 1 for nonzero a. So 0^0 can't continue both of these patterns by being both 0 and 1.

One of the nice reasons to choose 1 over 0 for 0^0 is to make the statement "the cardinality of the set of functions from a size n set to a size m set is m^n". There's exactly one function from the empty set to itself.

hackerbrother• 7 years ago

It's just a definition!

mikorym• 7 years ago

As the article says, one could consider this a matter of definition. The definition of choice for me (and some other category theorists?) is that the prime numbers are the second row in the devisibility lattice. (That is, n < m iff n | m.)

Then, the number 1 is the single element of the first row and all the prime numbers are the atoms immediately above 1.

waffleguy• 7 years ago

honestly this is where math stops being logical and follows emotions. The logical reason one isn’t prime is because we don’t want to ruin theorems that were based on faulty logic so we continue with the faulty logic... yay?

FartyMcFarter• 7 years ago

Short answer: 1 could be counted as a prime number, and in fact it used to be counted as a prime number by some mathematicians [1]. These days, most people just choose not to do that, since this makes most proofs and definitions more convenient.

Shorter answer: Convenience.

[1] http://mathworld.wolfram.com/PrimeNumber.html (3rd paragraph)

arithma• 7 years ago

I am not a mathematician.

I reached solace regarding this itchy issue (not even joking) by considering the following: Primes are the minimal set of numbers to generate another set of numbers, using multiplication (with repetition, yada yada) For the positive numbers, [2, 3, 5, 7, ...] is all we need, considering that, as many others have said, 1 = Product([]).

In that sense, to generate all integers, the integers you need are: [-1, 0, 2, 3, 5, 7..]

Similarly, to generate all gaussian integers (complex numbers with integer components,) you can follow the following link: https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_prim... however, am not sure why 0 is not considered a prime number in this context.

Gaussian primes are symmetric over the real and imaginary axes, and to me it's quite curious why 0 is not considered prime there, but I guess uniqueness of the multiplication matters too (0 can be repeated multiple times, which could make some theorems about uniqueness of factorization a bit cumbersome as well.)

rocqua4• 7 years ago

I am not sure why 0 is not considered a prime number in this context.

So, if you have a set with multiplication and addition, that is called a ring. A very nice class of rings is called fields. This is where division is possible. Another way to state that 'division is possible' is to say that multiplication is a group operator on the entire set except zero.

contravariant• 7 years ago

You might be happy to know that 0 is considered a prime element in typical cases (barring those cases where there are 0 divisors, i.e. where two elements can be multiplied to create 0, obviously this is not an issue with regular numbers).

Edit: for historical reasons prime elements are slightly different from the usual prime numbers, and 0 remains a bit of a special case even now.

yakubin3• 7 years ago

I am not sure why 0 is not considered a prime number in this context.

The greatest common divisor of any two primes is one. To say that two numbers are coprime, is to say their greatest common divisor is 1. If we considered 0 a prime, it would be a bit weird (and messy) that two prime (0 and any other prime number) numbers aren't coprime.

Why isn't 1 a prime number?

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