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Integer Complexity

Ed Pegg
Organization: Wolfram Research, Inc.
Department: Scientific Information Group
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Using the set of symbols {1, , +} and parentheses, how many 1's does it take to represent 23? It turns out that eleven are needed. Surprisingly, the number 27 requires only nine.

23 = (1+1)((1+1)((1+1)(1+1)+1)+1)+1
27 = (1+1+1) (1+1+1) (1+1+1)

This notebook provides code to evaluate many different types of integer complexity. In the PlusMinus section is code that was used to disprove a conjecture listed in the Encyclopedia of Integer Sequences.
A Math Games column talks more about the topic. Also, Stephen Wolfram's NKS Online, Page 916, discusses it.

*Mathematics > Recreational Mathematics

Operator functions, complexity, plus times problem

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IntegerComplexity.nb (18.2 KB) - Mathematica Notebook [for Mathematica 5.0]