Teaching a lot of developmental classes, I spend a lot of time with basic concepts like the fundamental theorem of arithmetic. If you are as geeky as me, it charges your card to no end that every natural number can be written uniquely as a product of primes. It is so strange (and wonderful) that a certain subset of numbers, the primes, are the multiplicative backbone of all the natural numbers. I think of the fundamental theorem as a decompostion theorem in the sense that shows us that all the naturals can be broken down in a particular way. Another of these “decompostion” theorems is Lagrange’s four square theorem, that any number can be expressed as the sum of four or less perfect squares. Why this theorem is cool but not as cool as the fundamental theorem is that the decompostion of a number in squares is non-unique. Think 64 = 8^2 and 64=4^2+4^2+4^2+4^2. Also Lagrange’s decompostion is a decomposition in two operations addition and exponentiation. So the fundamental theorem is one of many, but also special.

Lately, I have been wondering what “prime” would be for operations other than multiplication? This is what I was thinking about: a natural number *n* is prime in operation *@* if the only way to decompose *n* in *@* for the naturals is *n**@**e*, where *e* is the identity for *@*. Applying this to addition there is only one prime number, 1. Every other number is composite. Take 4, it can be written as 3+1, 2+2, or 1+1+1+1. The latter is in fact the additive prime factorization of 4. Addition is kind of boring though. Not enough primes.

Note, I left the identity on the right-side in the definition, so that we could apply the definition to exponentiation. Here we have a lot more primes. In fact the only composites are the perfect squares, perfect cubes, perfect….. The first ten exponential primes are 2, 3, 5, 6, 7, 10, 11, 12, 13, 14. Compare that to the first ten multiplicative primes 1, 2, 3, 5, 7, 11, 13, 17, 19, 23. We can’t really say that there are more exponential primes than multiplicative primes. (Showing that there are an infinite number of exponential primes is, I think, just a matter of adapting the same old proof we had in multiplication.) What we can say is that the exponential primes definitely seem more dense. In fact, I think it stands to reason that (other than 1) every multiplicative prime is by necessity an exponential prime. Now imagine primes for super/hyper-exponentiation. The primes get more and more dense.

Now let’s play. What is the exponential prime factorization of 16?

What about the hyper-exponential prime factorization of 16?

pic by Adrian Leatherland