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So if I wanted to send 1 satoshi from a Bitcoin address to the same address...

Would this be possible?

Would the transaction be recorded on the blockchain?

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    Yes, this is possible.

    As far as bitcoin is concerned there is no such thing as an address, only locking scripts. As long as the input's lock script is satisfied, the output can be sent anywhere, including to an identical script.

    This technique is often used by exchanges to consolidate the smaller outputs that build up as they process withdrawals into more usable chunks when fees are low.

    • Also used for change in some wallets. – JBaczuk Aug 17 '18 at 13:48

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