Very Compact Expressions for Amplitudes

A number theoretic algorithm is given for writing gauge theory amplitudes in a compact manner. It is possible to write down all details of the complete $L$ loop amplitude with two integers, or a complex integer. However, a more symmetric notation requires more integers, five or seven, depending on the type of theory. It is possible that in the symmetric form (or in the non-symmetric form) that a direct (or less direct) recursive algorithm or generating function can be developed to compute these numbers at arbitrary loop order. The existence of this function is implied by the recursive structure of loop amplitudes and their analyticity, i.e. multi-particle poles; a function requiring a finite number of computations such as a polynomial with derivable coefficients is desired.