## And I wonder…

Start with the operation addition. We define an operation multiplication by iterating addition. For example, 2*3 is defined as 2+2+2. And in general,

$a*b=\begin{matrix} \underbrace{a+a+...a}\\ b\, times \end{matrix}$
Likewise, we can form exponentiation by iterating multiplication. E.g. 2^3 is defined as 2*2*2. And in general,
$a^b=\begin{matrix} \underbrace{a*a*...a}\\ b\, times \end{matrix}$
Addition, Multiplication, and Exponentiation are operations we are very familiar, but it isn’t hard to see that this chain of operations can be extended infinitely. The next operation, often called tetration, can be defined as a^^b = a^a^a…a (where a is being raised to itself b-many times) The whole sequence of operation is sometimes called the hyperoperation sequence.

All of this is a long preamble to a question. Once you know the facts of addition, you can gain any fact of multiplication or a fact of any subsequent operation down the hyperoperation chain. However, is it possible to discover the facts of addition from the facts of multiplication by reversing the iteration? For example, knowing “3*2=6″ gives me the addition fact “3+3=6″, but can I get all the addition facts?