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I was wondering what the shortest way to represent any given number is. For example, $387420489=9^9$. So, for this case, the smallest representation is of order 2 (2 numbers). Alternatively, $10=2\cdot5$ also has 2, but it began with 2, so there is nothing constructive there.

The symbols permitted in the expression are $+,-,\times$ and taking exponential is also permitted.

Edit: To make this into a question, is there a general form to numbers that have a least character representation that is less than their natural representation?

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    To make this interesting you should allow division and bracketing, and only allow the digit 1. In that case I think that the expansion of the prime factorization may be the shortest expression.2013-02-15

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