Why 1,0,and 6 are prime (in a manner of speaking…)

This is a fun little tidbit that may not ever be worth a video, since it’s rather silly. To start with, let me be clear that I’m not saying that any of these three numbers are prime in the usual sense. (Many mathematicians used to count 1 as prime, but it has become clear that that is a bad definition; it messes up the fundamental theorem of arithmetic, for example.)

So what is special about an ordinary prime number? First, p is prime if it cannot be created by multiplying together positive integers, none of which is equal to p. So if you’re making all the positive integers and using only multiplication, you must include all primes in your recipe—they are necessary ingredients. Second, any positive integer can be so created—they are sufficient ingredients.

OK, so in what sense is 1 prime? Just change multiplication in the paragraph above to addition. Is there some positive integer that cannot be created by adding together other, more basic positive integers? You bet: the number 1. Can any integer be created by adding a bunch of 1’s? Yes indeed. So 1 is the unique “prime” for the addition operation. Note how different this is from multiplication, which requires an infinite number of building blocks. Addition is (no surprise) much simpler.

What about 0? First, let’s expand a tiny bit to considering the natural numbers (the nonnegative integers), so that 0 is actually in the mix. Next, just as multiplication is built by doing repeated addition, addition is built out of a simpler, more basic operation, the successor operation. (I say a lot about this in my videos about infinity and ridiculously huge numbers.) The successor of any number is simply the next number, one bigger. The famous Peano axioms  are built around starting with 0 and the successor operation, and are enough to build much of mathematics. The Peano axioms essentially say that:

  1. 0 is not the successor of any natural number.
  2. Any natural number can be obtained by starting with 0 and repeatedly applying the successor operation.

That may sound familiar—it says that 0 has exactly the properties of being a “prime” for the successor operation. It is the (unique) seed that you need to put into the process to create everything else.

(In fact, if you go all the way back to set theory and Von Neumann’s construction of the natural numbers, you realize that zero is none other than the empty set, the ultimate irreducible object. Von Neumann essentially created all of mathematics from nothing. Again, see my videos on infinity.)

OK, but what about 6? That’s something we get if we go the other direction on the ladder of operations, starting from multiplication. We could consider exponentiation of positive integers, m^n, and imagine making all positive integers by starting with a seed set (the “primes”) and allowing ourselves only to use exponentiation. What will those “primes” be, i.e. what numbers cannot be written in that form? Well, ordinary primes cannot, so we’ll need those as seeds. Conveniently, once we have those, then any desired exponent can be obtained, since m^{kl} = (m^k)^l. Hence we’ll get all perfect powers of primes. But try as we might we will not get 6. So in this (strange) sense, 6 is prime. 

OK, so that was mostly silly, but it’s actually good practice for thinking about the very general notion of building up complicated objects from simpler, ideally irreducible, building blocks. That applies in many, many contexts beyond arithmetic, in lattice theory, group theory, representation theory, category theory, logic, you name it.

 

 

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