  This paper studies the minimal length representation of the natural numbers.
Let O be a fixed set of integer-valued functions (primarily hyperoperations).
For each n, what is the shortest way of expressing n as a combinations of
functions in O to the constant 1? For example, if O contains the two functions
Sx (successor of x) and *xy (x times y) then the shortest representation of 12
is *SSS1SS1, with 8 symbols. This is taken to mean that 8 is complexity of 12
under O. We make a study of such minimal representations and complexities in
this paper, proving and/or rightly predicting bounds on complexities,
discussing some relevant patterns in the complexities and minimal
representations of the natural numbers and listing the results gleaned from
computational analysis. Computationally, the first 4.5 million natural numbers
were probed to verify our mathematically obtained results. Due to the
finiteness of the problem, we used the method of exhaustion of possibilities to
state some other results as well.
