This whole post is a reply to someone on Twitter; who has asked what I think is an excellently conceived question because what often appears to be the simplest of things often turns out to be elegantly intricate.
"Okay, you're a smart guy. Explain to me why 1 isn't prime."
This question gets at one of the things in mathematics that everyone takes for granted but when thought about for long periods of time, is like an eel on the end of a fishing line and gets tied up in knots needlessly.
To answer this, we'll come at the problem backwards and eliminate that which doesn't apply because whatever is left, however improbable, must be the truth.
To begin with, we are concerned only about integers; which are rational, real, and positive. We don't want to concern ourselves with fractions or negative numbers here.
A Composite number is one which is composed of many factors.
12 can be divided into 6 parts, 4 parts, 3 parts, 2 parts, and left alone as one part; all arrive at whole numbers.
A Prime number only has two factors; which are itself and 1.
7 can only be divided into 7 parts.
A Prime number n, can only be divided into n parts nicely. If you divide Prime numbers into any other number of parts, you get weird bits.
Imagine that we are in charge of Horse Chocolate Factory. We make a bunch of different lines of chocolate but we are best known for our blocks of chocolate.
Composite numbers are like blocks of chocolate that you can divide into rows, columns, or perhaps groups of identical smaller block sections.
Prime numbers are like blocks of chocolate which are in bars; which you can only break into individual squares and still get all of the bits be identical.
Now we know about Composite numbers and Prime numbers; so we're done, right?
No!
We're still left with the problem of 1. There is no meaningful way to break one square of chocolate into any number of pieces and while you could possibly say that it fits the rule for Prime numbers that you can divide it into 1 part nicely and still get 1, without any weird bits, you haven't actually divided it, have you?
1 appears to be a special case which doesn't fit into the definition of being Composite or being Prime, without us looking like a fool.
1 is a special case because 1 can not be divided meaningfully. Dividing 1 by 1 to get 1 sounds like it should be a trivial case except for the fact that it is a unique case. If 1 is 1 and all alone and evermore shall be so, and 1 is the loneliest number that you will ever find, then it's practically standing here screaming out for it's own unique classification in it's own unique voice.
So we give it one.
1 is unitary.
There are a class of these special numbers. The units, or the divisors of unity, are the basic elements of mathematics which have a multiplicative inverse. It probably goes without saying that 1 has an evil twin in -1 and I guess that if we're going to expand our definition to include the Gaussian integers, then we have four divisors of unity which are 1, -1, i, and -i.
1 isn't prime; because as far as the ontological question of how number are classified into basic categories and which exist on the most fundamental level, then we need to be extremely careful with 1 and with primes generally.
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