An encoding of large numbers

[Heiner Marxen]

 I'm not sure, whether this is really helpful, but just in case... I happened to find this one: Efficient Data Structure and Algorithms for Sparse Integers, Sets and Predicates by Jean E. Vuillemin.

Zero and 1 are atoms, and all other numbers can uniquely be encoded as triples n = T(g,p,d) = g + d*2^2^p, with g,d < 2^2^p and d>0.  All three, g, d and especially p are encoded the same.

Also noticable is the last sentence: We thank Don Knuth for his truly constructive criti-
cism of an early draft.

[Shawn Ligocki]

Thanks Heiner, this is very interesting. I especially like how they have made a standard way to represent every number and how it is efficient to compare two numbers. This is not (yet) true for the system I am using. But it seems that many numbers we need to represent would be sense in this notation (like 3^^n ex I assume).

FWIW, my notation system is:

n = (c + sum(a_i * p^m_i))/d

where c,d, and all a_i are basic integers and m_i are either integers or recursively defined in the notation again. I have a little bit of standardization, but even testing equality of two numbers in this notation is not obvious. If you restrict to a single a * p^m term I think a number of problems would become easier, but Pavel has found a 2x6 TM which works not be simulatable with that simplified notation.

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