Yeah that comment didn’t make much sense. No one would seriously dispute that 2 is the first prime.
Some ancient mathematicians such as Aristotle and Plato solved that problem by saying 2 is the first number. Others stated 3 to be the first prime
That isn’t holdable once you accept 0 and negative integers to be numbers.
https://cs.uwaterloo.ca/journals/JIS/VOL15/Caldwell2/cald6.h... states that, for example, Goldbach thought 1 to be prime at some time (in a letter to Euler), as did Legendre, Lebesgue (sometimes), Cayley, Kronecker, Hardy, Lehmer (as the article discussed also says), and the aliens in Carl Sagan’s “Contact”.
It also gives fairly recent publications that state 1 is a prime.
In the end, whether we consider 1 to be prime is more a choice (just as mathematicians commonly chose to pick 0⁰ = 1) than that it necessarily the case. It just is the better choice (https://en.wikipedia.org/wiki/Prime_number#Primality_of_one)
I'm not sure how we go to this point exactly but I've drunk a lot of wine and will now take stage left.
No one would seriously say the latter.
Given that a counting always starts at one, because that is what defines counting, then there cannot be a zeroth prime.
You can define counting as defining an injective function from your set to the natural numbers, and then you need to have some element going to 0 - as per the definition of a countable set[0].
Also, both the cardinal[1] and the ordinal[2] numbers are defined as starting from 0, just like the cardinal numbers.
[0] https://en.wikipedia.org/wiki/Countable_set
You can define counting as an injection into other (equivalent) sets just as rigorously.
And it's not even universally agreed on that the natural numbers should include 0. Wikipedia mentions the different conventions: https://en.wikipedia.org/wiki/Natural_number
(I like my natural numbers to start with 0. But that's just because 0 is my second most favourite number. Starting with 1 is legitimate.)
The process of counting might be defined as what starts to happen when you stick up one finger and say something to emphasise what that finger means. That something will not be zero. Ever.
When you count your sheep into your pen, you will of course start: "Yan, tan, tither, toe" Trust me that yan does not mean zero.
Also note that if you search those terms, you will get a valid result and conclude I've misspelt some of those Cumbric words. I haven't, according to living relos of mine. Speling is a bit odd anyway when you go back a few centuries and I'll wager that tither is more likely than tethera because it is very slightly more easy to say. Tethera is three syllables but tether is two, bordering on one. However tethera could be pronounced "tethra", ie drop the extra e when spoken.
The counting numbers are fairly rigorously defined and are the numbers we use when we don't have access to more than the usual four dimensions, imaginary thingies, quaternions, etc etc.
The counting numbers start at one (probably)
More seriously
http://mathforum.org/library/drmath/view/55958.html
But the link to details is dead.
https://groups.google.com/d/topic/geometry.research/7pyFhAAy...
I saw that, looking for references... very amusing. Even funnier (to me):
https://www.google.com/search?q=is+1000000101110000000000000...
You can certainly give a name to the integers {-1} U P, but maybe it would be better to call them "choice" or "select" numbers.
> Every nonzero rational number has a unique factorization into powers of distinct primes.
As you note, (-1)^2 = 1. But if you read carefully, you'll see that the factorization of -100/3 is uniquely:
-1 * 4 * 1/3 * 5 * 1 ...
1 * 4 * 1/3 * 5 * 1 ...
And yes, changing the definition of "prime" does change some special cases -- it removes some (such as extending unique factorization to negative rationals), and adds others (wherever primes are assumed positive).
[1] http://swc-alpha.math.arizona.edu/video/2009/2009ConwayLectu... (mention around 7:00)
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