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It’s not impossible, it’s just that the number space is so enormous that it’s exceedingly unlikely that it will ever be found: every random private key you generate will map to one of 2160 possible P2PKH addresses.
Also see, Bitcoin Stack Exchange Is each Bitcoin address unique?
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Exactly there is no "burn address" built into bitcoin code, but there are addresses that are so improbable that there is no real way someone holds the private key.
There are tools that generate "vanity addresses" (along with private key) but typically it becomes impossible after the first few chars. So you can probably easily generate private key for bc111XXXXX address but generating bc111111111111111... is impossible.