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According to the https://en.bitcoin.it/wiki/Address site, it states that "most" addresses have a private key. Here is the relavent text:

Most addresses have a "private key"

For most addresses, there is a corresponding secret number known as a private key that is required to spend funds previously sent to an address. When using a Bitcoin client, private keys are typically stored in the wallet file. The private key has a special purpose - it is mathematically needed to create valid transactions that spend the funds originally sent to the address. If the private key to an address is lost (for example, in a hard drive crash, fire or other natural disaster), any associated Bitcoins are effectively lost forever.

I have always assumed that Bitcoin worked in such a way that all addresses must have a matching private key due to the public/private key encryption on which it is based. In what cases would a bitcoin address not have a private key?

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If an address merely 'looks like' an address and all other aspects of a transaction are ok then miners will include it in a block.

This has been (ab)used to embed ASCII art into the blockchain at the cost of permanently destroying some bitcoin.

For an example with explanation see https://bitcointalk.org/index.php?topic=33618.msg420597#msg420597

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    Here's another example: 1BitcoinEaterAddressDontSendf59kuE Most certainly nobody has the private key that results in that Bicoin address (it would take more than trillions of years of all of today's existing computing capacity for a VanityGen type of app to find a private key that would hash to that Bitcoin address. – Stephen Gornick Sep 12 '12 at 20:52
  • And a related forum thread: bitcointalk.org/index.php?topic=26539.0 – Stephen Gornick Sep 12 '12 at 20:52
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    So the issue isn't so much that a private key doesn't exist as that no one knows (or can know) what the corresponding private key is... got it. – Jason Southwell Sep 13 '12 at 4:32
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    @JasonSouthwell: And it is at least theoretically possible that there might not be one. – David Schwartz Sep 14 '12 at 19:50

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