There were people claiming that he is or they are Satoshi, but I know that there is only one way to prove if the person is saying the truth.
There is an account here in Bitcointalk.org which he/they used long time ago.
The third person who created an account here is the real Nakamoto. : https://bitcointalksearch.org/user/satoshi-3
One message using that account supporting who is the person or group behind this will end the conversation.
That's not proof. First, the account is locked by admin (theymos) and the only way to have it unlocked is to prove to theymos they are Satoshi. That proof is either through a signed bitcoin message from either the genesis block or several of the earliest blocks that we know were mined by Satoshi, OR a GPG signed message.
Preferably, both a bitcoin signed message AND a GPG signed message.
We have his GPG public key, and we have the first thousand blocks, so we have a thousand addresses, if he can just sign from at least 3 of them (the more the merrier, but that gets tedious) then we have a very high degree of proof, that whoever did that has access to both the wallets and coins and the GPG private key.
So far, no one has done it, and it's likely no one else can do it.
All the fake satoshis who've tried, can't do it, not because they don't want to, but because they are incapable of it.