On the Arithmetic of Recursively Run-length Compressed Natural Numbers

We study arithmetic properties of a new treebased number representation, Recursively run-length compressed natural numbers, defined by applying recursively a run-length encoding of their binary digits. Our representation supports novel algorithms that, in the best case, collapse the complexity of various computations by superexponential factors and in the worse case are within a constant factor of their traditional counterparts. As a result, it opens the door to a new world, where arithmetic operations are limited by the structural complexity of their operands, rather than their bitsizes. Keywords-run-length compressed numbers, hereditary numbering systems, arithmetic algorithms for giant numbers, structural complexity of natural numbers