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I've noticed several addresses receive AND SEND tokens in Ethereum which have an uncomfortably large number of leading 0s.

[1] I thought all transfers had to be signed by a private key that corresponded to the Ethereum address owning those tokens? That's why I'm not sure how this happens.

I'm sure I'm missing something about my understanding of ETH, I thought all addresses that could spend their ERC20 token balance, had to have a valid cryptographic signature coming from a private key that corresponded to their the public address.

[2] Could someone give me a brief explanation of why this address is able to send the ERC20 token NMR, out of its balance?

Address in question: 0x0000000000000000000000000000000000028Ae8

Token transactions: https://etherscan.io/address/0x0000000000000000000000000000000000028Ae8#tokentxns

[3] Additionally, I would like to know what is the longest vanity address that has been found to exist which was generated via brute force, or a clever vanity address finding algorithm.

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The example you provided doesn't show an address with 35 leading zeros sending a transaction, but sending and receiving ERC20 tokens.

Ad 1: You're right in the first part, but ERC20 transfers are not the same as Ethereum transactions. The former are so to speak "internal transactions" inside of the latter, where storage variables inside of the contract's mappings get updated. (However, by sheer luck someone theoretically could just stumble upon the magic seed words that create an address with 35 leading zero in any case.)

Ad 2: This is where you're mistaken: your link points to the Erc20 Token Txns tab on Etherscan. The "from"-address we see here doesn't show the sender of the Ethereum transaction, but the "from"-value of the ERC20 Transfer event that the smart contract emitted. Apparently the Numeraire ERC20 contract emits Transfer events with such "from" and "to" values. (I don't know why it does that, I'm not familiar with that contract).

Ad 3: I don't know what the lowest Ethereum account address is, but 0x000000000000d4f126f9a0339FE7a77B07d3F2C2 for example is an account with 12 leading zeros. A Reddit user claims to have created it in 9h recently (on a RTX 3060). Another kind of example is 0x000000000000006F6502B7F2bbaC8C30A3f67E9a, which is a contract address with 14 leading zeros (spotted on Reddit about 2 years ago).

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    Thank you, this is the most complete answer, since most tokens require the signature from the address holding them , i got fooled by this distinctive contract Jun 23, 2022 at 21:47

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