user11212: The Base58 Encoding of a public key hash outputs:  "A set of 58 alphanumeric symbols consisting of easily distinguished uppercase and lowercase letters (0OIl are not used)"
This set is a fixed string of 34 characters.
That is to say, "A bit coin wallet's address is 34 characters, each character being one from a possible set of 58"
So the total possible addresses is 58^34 which is just slightly less than 2^40.
I understand that the hash space is 2^160, but the final result is much smaller at 2^40.  What am I missing?

Edit:  I made a simple math mistake reducing 58^34 to a power of two
  by adding not multiplying the exponents...  The bitcoin address is
  sufficient to contain the 2^160 possible hashes:  58^34 is 904
  octodecillion (60 decimal digits), and 2^160 is 1 quindecillion (49
  decimal digits)... the larger number accounts for the version byte and
  checksum.


user9175: No, 58^34 is greater than 2^160: http://www.wolframalpha.com/input/?i=2*160+%3E+58*34
Note, that the address also contains a checksum and a network id not carrying additional information. That's why the numbers are not equal.

11212: <p>The Base58 Encoding of a public key hash outputs:  "A set of 58 alphanumeric symbols consisting of easily distinguished uppercase and lowercase letters (0OIl are not used)"</p>

<p>This set is a fixed string of 34 characters.</p>

<p><strong>That is to say, "A bit coin wallet's address is 34 characters, each character being one from a possible set of 58"</strong></p>

<p>So the total possible addresses is 58^34 which is just slightly less than 2^40.</p>

<p>I understand that the hash space is 2^160, but the final result is much smaller at 2^40.  What am I missing?</p>

<blockquote>
  <p>Edit:  I made a simple math mistake reducing 58^34 to a power of two
  by adding not multiplying the exponents...  The bitcoin address is
  sufficient to contain the 2^160 possible hashes:  58^34 is 904
  octodecillion (60 decimal digits), and 2^160 is 1 quindecillion (49
  decimal digits)... the larger number accounts for the version byte and
  checksum.</p>
</blockquote>
 9175: <p>No, 58^34 is greater than 2^160: <a href="http://www.wolframalpha.com/input/?i=2%2a%2a160%20%3E%2058%2a%2a34" rel="noreferrer">http://www.wolframalpha.com/input/?i=2*<em>160+%3E+58</em>*34</a></p>

<p>Note, that the address also contains a checksum and a network id not carrying additional information. That's why the numbers are not equal.</p>
