(Unrelated to pedagogy, but related to when multiplication is not repeated addition.)

When I was an early college student I started to become curious about the prime numbers. Why are some numbers in particular prime, and others aren't? I started with a thought experiment:

What if it were that 2*2 = 5 ?

Pretty quickly I realized that the relationship between addition and multiplication could not remain the same, because the distributive property guarantees that 2*2=4

  2*2 = (1+1) * (1+1) = 1+1+1+1 = 4

So I sought to define some form of multiplication that was as close to regular multiplication as possible, but without the distributive property. I ended up defining this generalized form of multiplication as a function m:NxN -> N (N = the natural numbers 1,2,3,4,...) with the following properties

1. Associative

2. Commutative

3. Multiplicative Identity

4. Increasing ( If i,j > 1, then m(i,j) > i,j )

5. Bigger number, bigger product ( If j > i, then m(j,k) > m(i,k) )

6. Uniqueness of Prime Factorization

An easy example is the odd numbers.

  1,2,3,4,5, ... n
  1,3,5,7,9, ... odd(n)

  odd(n) = 2n-1
  inverse_odd(x) = (x+1)/2
  m(i,j) = inverse_odd( odd(i) * odd(j) ) = 2ij - i - j + 1

Indeed, for the odd numbers m(2,2) = 5! The second odd number (3) multiplied with the second odd number (3) equals the fifth odd number (9).

It turns out that this definition is really a (nice) subset of something known as the Beurling Integers. The Beurling Integers are neat, because you can basically choose whatever distribution of prime factorizations you want (following the rules 4, 5, 6 above) and find a sequence of real numbers that satisfies that distribution. The catch is that we had to sever the ties of addition and multiplication.