Can anyone explain the mathematics that the article glosses over? What constitutes a valid bitcoin? Can I sit down and start generating 'em just in case?
It's using digital signatures to transfer coins. The prior owner will sign a coin and the new owner's public key. The new owner will verify the prior signature before accepting the coin. This is a longstanding idea going back to the 80's.
The rest of the details are to prevent double-spending without having a centrally trusted authority. I haven't spent time looking at this and can't say whether it works as advertised.
I don't know the maths behind it, but as far as I onow, yes you can go ahead and start generating them. You have to reverse some washes, ie it's CPU bound, which is how the scarcity comes in.
