There are also known addresses with no private keys.
There are no addresses with no private keys.
Better formulation: It hasn't been proved that there's an address that has no private keys. On average, each one has 2
96 different keys.
There are the "zero" addresses, formed from a private key with all zeroes. This is often seen in altcoins / shitcoins where they publicly burn coins.
There are also the vanity looking addresses such as 1BitcoinEater:
https://blockchain.info/address/1BitcoinEaterAddressDontSendf59kuEIt's never sent any coins, always just received them. Very unlikely anyone has the private key to that address, or might even be possibly no such private key can map to it. We do not know for sure, as like you said, it can not be proven. However, it is my opinion that the chances of anyone ever able to spend from that particular address is close to zero. Maybe 0.000000001% or something. Yes, there is a chance.
As far as I am concerned, it's impossible. Your definition of the word impossible may differ from mine. Normal usage of the word is usually hyperbole, but the meaning and intent is clear. It is impossible. Won't stop you from trying, but I'm sure you or anyone else will fail.
31h1vYVSYuKP6AhS86fbRdMw9XHieotbST
https://www.blockchain.com/btc/address/31h1vYVSYuKP6AhS86fbRdMw9XHieotbSTThis address is unspendable. And this time I mean it in the correct sense of the word. It's a hash from zeroes. You can see coins sent to it, but never spent.
Since addresses are encoded hashes, start with a hash, of which it would be nearly impossible to find the pre-image, e.g. assume a script hash of 0000000000000000000000000000000000000000. Encode that into a valid address:
bs58 -c 050000000000000000000000000000000000000000
31h1vYVSYuKP6AhS86fbRdMw9XHieotbST
Here is another address: 1CounterpartyXXXXXXXXXXXXXXXUWLpVr
Feel free to try and find out the private key to that one, as no one knows it. It was generated without knowing the private key, just an address. My use of the word impossible for this one just means very highly improbable.