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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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Although for any given number one can search, I don't see a way to answer the question in general. What are you looking for? – Ross Millikan Nov 17 '11 at 19:20
@Ross, is there any interest in this type of problem? I was looking for a general solution. – picakhu Nov 17 '11 at 19:48
Just out of curiosity, why are you not allowing division? – Robert Israel Nov 17 '11 at 20:17
What Ross has said. I note that the smallest (positive integer) number that has a representation shorter than just writing it in decimal is $125=5^3$, followed by $128=2^7$, $216=6^3$, $243=3^5$, $256=2^8=4^4$, $343=7^3$, $512=2^9=8^3$, $625=5^4$, $729=3^6=9^3$. – Gerry Myerson Nov 17 '11 at 23:13
The sequence of numbers @Gerry gave is in the OEIS. – J. M. Nov 18 '11 at 2:14

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