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Can we do arithmetic in a completely different way, with a radicallydifferent data structure? Could this approach provide practical benefits, likeoperations on giant numbers while having an average performance similar totraditional bitstring representations?  While answering these questions positively, our tree based representationdescribed in this paper comes with a few extra benefits: it compresses giantnumbers such that, for instance, the largest known prime number as well as itsrelated perfect number are represented as trees of small sizes. The same alsoapplies to Fermat numbers and important computations like exponentiation of twobecome constant time operations.  At the same time, succinct representations of sparse sets, multisets andsequences become possible through bijections to our tree-represented naturalnumbers.

Tree-based Arithmetic and Compressed Representations of Giant Numbers