
Why isn't 1 a prime number? - gpvos
https://blogs.scientificamerican.com/roots-of-unity/why-isnt-1-a-prime-number/
======
frankbreetz
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
That’s great. Thank you for sharing.

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

~~~
hackerman12345
Reminds me of The Riddler...

~~~
myself248
This is totally something that Monday and Frankly would've encountered on
Mathnet.

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earthicus
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](https://ncatlab.org/nlab/show/too+simple+to+be+simple)

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

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

~~~
earthicus
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.

------
bo1024
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
> 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](https://en.wikipedia.org/wiki/Idempotence)

~~~
bo1024
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.

------
Someone
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...](https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.pdf))

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.

~~~
nimih
> 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.

~~~
Someone
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”
    

to

    
    
      “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.

~~~
nimih
As far as I can tell, the idea of “same size means establishing a bijection”
appears (in a formal sense) at essentially the same time as it was discovered
that there are pairs of infinite collections where you can’t do that (i.e.
with Dedekind and Cantor in the late 1800s), so I don’t think that second
definition ever actually existed in a meaningful sense, although I think your
intuition seems reasonable (albeit “bending” holds the wrong connotation, imo,
as this would seem to be a rather principled generalization of an existing
concept rather than an arbitrary decision among a number of somewhat
reasonable options). Regardless, I’d be very interested to read a survey of
the history of ideas about cardinality/ordinality if you have a good one,
since modern treatments of these ideas are radically different than how humans
understood them even 150 years ago.

------
gabrielblack
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
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.

~~~
mintplant
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.

------
arithma
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...](https://en.wikipedia.org/wiki/Gaussian_integer#Gaussian_primes)
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.)

~~~
contravariant
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.

------
HenryBemis
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](https://en.wikipedia.org/wiki/Prime_number)

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

~~~
empath75
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.

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

~~~
ColinWright
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.

~~~
gerdesj
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.

~~~
ColinWright
> _Those PhD theses seem a little misguided (if well meaning) to me._

That's a strong statement to make as a throw-away. You haven't read them, you
haven't done the research, so I'd ask - what experience do you have of
tracking the effect of specific language on the acquisition and development of
mathematical skills in a significant sized population of very young children?

I suspect the answer is not a lot, and dismissing someone's research on the
basis of effectively no information is poor form.

------
mikorym
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.

------
dmarlow
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.

------
FartyMcFarter
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](http://mathworld.wolfram.com/PrimeNumber.html)
(3rd paragraph)

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

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

~~~
traderjane
I would argue that Zero is generally considered to be a natural. The naturals
are often constructed via an initial object, and you can call that initial
object whatever you want (some people write Z or Zed), but it's metaphorically
a zero.

~~~
rocqua
The pushback I have gotten om my tongue-in-cheek rants defending 0 as a
natural number would suggest not everyone agrees.

Some people really care about 1/n being possible for every natural number N.

~~~
yakubin
And other people care about natural numbers being the set of cardinal numbers
for finite sets. Others want the set of natural numbers with addition to be a
monoid. And actually most-known series in analysis are usually indexed
starting from zero, because that's when the formulas are usually simpler.

~~~
traderjane
But if you asked the same people to build the naturals, they'd likely include
an initial object that behaves like 0.

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

~~~
mcguire
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.)

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

------
pcmaffey
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.

------
hackerbrother
It's just a definition!

------
crazygringo
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.

~~~
fjsolwmv
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.

------
bensalem
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
Sure, but the next question—which the linked article is _about_ —is, why is
that the definition?

------
Hackbraten
And, even more important: is it Numberwang?

------
waffleguy
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?

------
HocusLocus
The number 1 is a prank by Loki.

------
blu42
"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
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.

~~~
blu42
Doh, my reading balked at the embedded add. Yes, p being a prime actually
inverts the statement successfully. My bad.

