
On Primes and Pluto - mjs
http://www.qedcat.com/archive/189.html
======
wbhart
For the ordinary natural numbers, 1 is the only number which you can invert,
i.e. 1/2, 1/3, 1/4, etc. are not integers, but 1/1 is. If we take in all the
integers, positive, negative and zero, then 1 and -1 are invertible. But
things become much more complicated in other number rings. For example, in the
number ring obtained by adjoining the square root of 7 to the integers (i.e.
Z[sqrt(7)]), there are infinitely many invertible values. For example 3
sqrt(7)-8 is invertible. Its inverse is -3 sqrt(7)-8. But any power of this
value is also invertible, so in fact there are infinitely many invertible
values in this ring. Anyway, the problem with units (invertible values) is
that they divide _any_ number in the number ring (after all, by definition,
they divide 1, and 1 divides everything). They therefore aren't very useful
for unique factorisation (aside: unique factorisation exists for elements of
Z[sqrt(7)], though not for every conceivable number ring). Because of this, it
is natural to exclude units from being primes.

~~~
Avitas
Interesting with a side note about toying with Excel.

When I plug in:

    
    
        1/(3 sqrt(7)-8)
    
        and
    
        -3 sqrt(7)-8
    

...into Excel, there must be a rounding error. They resolve to:

    
    
        -15.9372539331939000
    
        and
    
        -15.9372539331938000
    

...respectively.

Time to plug these into Wolfram. I get:

    
    
        -15.93725393319377177150484726091778127713077754924735054110500337760320646969085088328117865942363083184519373501549238...
    
        and
    
        -15.93725393319377177150484726091778127713077754924735054110500337760320646969085088328117865942363083184519373501549238...
    

...respectively.

~~~
hdevalence
If you want to do exact computations, SAGE is handy, Free, and built on
Python:

    
    
        sage: k.<a> = QuadraticField(7); k
        Number Field in a with defining polynomial x^2 - 7
        sage: x = 3*a - 8
        sage: x.is_integral()
        True
        sage: 1/x
        -3*a - 8
        sage: x * (1/x)
        1
        sage: (1/x).is_integral()
        True
    

(Note: the is_integral() is because we define k to be QQ(sqrt(7)), not
ZZ(sqrt(7)), so we want to check that our elements are really integers, i.e.,
lie in ZZ(sqrt(7))).

------
tolmasky
Wow, so many comments from people that clearly didn't bother reading the
article (all the way through). I suppose its understandable, I too had to
fight the temptation to hit the back button and angrily type out "its just a
definition!". Luckily I decided to actually read the whole thing through,
notice that the author is quite aware of this, and be treated to a really
interesting history of math.

~~~
bdhe
Agreed. It is an interesting article to read. The survey cited is also
interesting in its own right [1]. We often forget that ideas we find trivial
and intuitive are only so because it took the work of many more people and
centuries to hone their ideas which at that time would be have been new and
ambiguous. This is especially true with mathematics because a lot of the field
focuses on getting the right abstractions to complex ideas which when done
correctly seem "trivial, natural, and straightforward." Also see the history
of calculus [2].

[1]
[https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.p...](https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell1/cald5.pdf)

[2]
[http://en.wikipedia.org/wiki/History_of_calculus](http://en.wikipedia.org/wiki/History_of_calculus)

------
i_c_b
My favorite "1 not behaving as a prime" example.

Here are two simple javascript functions:

function f(n,k){ var t = 0; for( var j = 2; j <= n; j++ )t += 1/k - f(n/j,
k+1); return t;} function p(n){ return f(n,1)-f(n-1,1); }

If you call p(n) when n is prime, it will return 1. If you call p(n) when n is
a prime power (so, say, 4 or 9 or 16), it will return 1/power (so p(4) is .5,
p(8) is .3333..., etc). If you call p(n) with a number with multiple prime
bases (so 6 or 14 or 30 or...), it will return 0.

And if you call p(1), it will return 0, NOT 1.

In fact, f(n,1) here is a compact (and slow) way of computing the Riemann
Prime Counting function:
[http://mathworld.wolfram.com/RiemannPrimeCountingFunction.ht...](http://mathworld.wolfram.com/RiemannPrimeCountingFunction.html)

Another way to compute this exact same function (given as (8) on that link)
uses the famous Riemann Zeta function zeroes, although that is much harder to
follow.

Now, the behavior of the Riemann Prime Counting function doesn't PROVE that 1
isn't a prime, which, as noted, is a question about definition. But what it
does do is show that, in an extremely important context, a context that seems
to be, mathematically, solely about identifying primes, 1 isn't behaving like
the primes at all.

------
minikites
There's a great Numberphile video that walks you through a similar line of
explanation:
[http://www.youtube.com/watch?v=IQofiPqhJ_s](http://www.youtube.com/watch?v=IQofiPqhJ_s)

------
sp332
Seems like the Law of Small Numbers
[https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers](https://en.wikipedia.org/wiki/Strong_Law_of_Small_Numbers)

------
fennecfoxen
At some level you really shouldn't care why 1 isn't prime. It's not-prime
because we defined it that way. 1 has some mathematical properties in common
with the numbers we happen to call "prime numbers" and some mathematical
properties which are different; in another world we could have called it prime
and replaced "the prime numbers" with "the prime numbers greater than 1" in
assorted theorems and EVERYTHING ABOUT MATH WOULD BE THE SAME.

~~~
cobbal
Well, yes and no. If you're just talking about integer primes then sure. If
you consider gaussian integers though (a + b*i where a and b are integers)
then you would have to say "the gaussian prime numbers that aren't 1, -1, i,
or -i.

The definition of primes that excludes units makes sense in more places than
one that includes units (especially once you start talking about prime ideals
instead of prime numbers)

------
shittyanalogy
Linguistically:

 _Prime_ is a word not a fact of the universe, just like _Planet_ is a word
and not a fact of the universe. Those words, unlike most words, have
definitions put in place by authorities. The authorities might change the
definitions over time as a new definition's usefulness exceeds that of an old
definition.

Practically:

It's more useful to think of primes and planets as not including their recent
ex-members than it is to include them.

~~~
diminoten
Yeah, the article covers that pretty early on. Turns out that's actually not
why 1 hasn't been prime for most of human history, though.

Actually read the article to find the answer!

~~~
shittyanalogy
I did read the article.

------
benched
The problem of 1 not being prime could be easily solved by having a separate
term, say "noncomposite" or whatever you like, that means the set of the
primes and one.

------
4shadow
If you let 1 be prime then you would violate the uniqueness property of the
fundamental theorem of arithmetic

------
deletes
It is not a prime because every integer greater than 1, must be made by a
unique product of prime numbers, or is a prime itself. If 1 were prime, that
would not hold true.

proof:
[http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmet...](http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic)

 _edited_

~~~
CatMtKing
What about prime integers? Aren't they a product of 1 and themselves?

~~~
happyscrappy
FTA:

What if we permitted 1 to be prime? In that case, 84 would also have the
"prime" factorisation 1 x 1 x 1 x 2 x 2 x 3 x 7. That is, 84 could still be
factorised, but it would no longer have a unique prime factorisation.

~~~
CatMtKing
I did read the article. I was just pointing out that the poster's original
statement seemed to exclude prime integers -- well, unless the empty product
is a product of primes.

------
LanceH
It isn't prime by definition.

The definition is made for convenience sake based on what it implies. One
being prime implies that other primes are evenly divisible by another prime
but for some reason that doesn't stop them from being prime. So 1's primeness
would have to be special relative to the primality of the other numbers.

Starting the primes at 2 makes all the definitions and implications simple,
except for the one caveat that primes start at 2.

Either way you'll have to make a concession, we choose the one that only has
to make it once.

The other way around you end up with things like: The square of a natural
number is not prime (except 1). The product of two primes is not prime (unless
one of them is 1). and so on...

~~~
imadethis
Yes it isn't prime by definition, thus the article.

