user2156: After the 21 millionth bitcoin or 20,999,999th bitcoin is mined, what will happen to bitcoin mining? - will it just stop, or will we make a further division of the satoshi thanks to Bitcoin's infinite divisibility? Or maybe miners could mine a share of the Bitcoin transaction fee...
Any suggestions/mathematical answers are appreciated.

user64908: miners will depend on transaction fees.. a block will continue to be mined every 10 mins. 
Though bitcoin will not live till that day. For bitcoin to live till that day, it will have to be worth more than a million dollar each. If it isnt that much worth, it will be expensive to mine

user5406: Pretty much the same will happen as with a Halving now: The only thing that changes is how many bitcoins may be created with the Coinbase transaction; except this time it drops to Zero instead of halving.
In particular, mining would have to continue in order to validate transactions, although it would then be completely funded by transaction fees. However, that will only be marginally different from the years before the last new coins, as it will will have been only satoshis per block for many years before that.

user2199: The short anwser is that miners will rely on transaction fees, to get paid.

user208: Every block has exactly one "coinbase transaction", the one transaction which doesn't have actual inputs, but gets all the fees and mining subsidy.
Every 210000 blocks, this subsidy halves. Right now, each block is allowed (not required!) to bring 50.00000000 BTC into circulation. Very soon, this will become 25.00000000. Four years later, 12.50000000. And this will continue: 6.25, 3.125, 1.5625, and so on. There are only 8 decimals though, and at some point the subsidy will only be 0.00000001. Four years after that, it will effectively become zero. This will not happen this century, though.
This doesn't mean that there won't be any blocks produced anymore. Assuming Bitcoin is still alive, we'll still have blocks being produced every 10 minutes. Miners will just get all their income from fees, without any subsidy.

user516: Assuming Bitcoin is still active at that point in time, mining will continue, because transaction fees will make it worthwhile to do so.
This topic has been discussed heavily in other answers, including:

What happens once the mining reward gets cut in half?
How many bitcoins will there eventually be?
How much will transaction fees eventually be?

The last link has suggestions that the fee mechanism of Bitcoin may need changing, though that's yet to be proven whether or not it will be a real problem.

2156: <p>After the 21 millionth bitcoin or 20,999,999th bitcoin is mined, what will happen to bitcoin mining? - will it just stop, or will we make a further division of the satoshi thanks to Bitcoin's infinite divisibility? Or maybe miners could mine a share of the Bitcoin transaction fee...</p>

<p>Any suggestions/mathematical answers are appreciated.</p>
 64908: <p>miners will depend on transaction fees.. a block will continue to be mined every 10 mins. </p>

<p>Though bitcoin will not live till that day. For bitcoin to live till that day, it will have to be worth more than a million dollar each. If it isnt that much worth, it will be expensive to mine</p>
 5406: <p>Pretty much the same will happen as with a Halving now: The only thing that changes is how many bitcoins may be created with the Coinbase transaction; except this time it drops to Zero instead of halving.</p>

<p>In particular, mining would have to continue in order to validate transactions, although it would then be completely funded by transaction fees. However, that will only be marginally different from the years before the last new coins, as it will will have been only satoshis per block for many years before that.</p>
 2199: <p>The short anwser is that miners will rely on transaction fees, to get paid.</p>
 208: <p>Every block has exactly one "coinbase transaction", the one transaction which doesn't have actual inputs, but gets all the fees and mining subsidy.</p>

<p>Every 210000 blocks, this subsidy halves. Right now, each block is allowed (not required!) to bring 50.00000000 BTC into circulation. Very soon, this will become 25.00000000. Four years later, 12.50000000. And this will continue: 6.25, 3.125, 1.5625, and so on. There are only 8 decimals though, and at some point the subsidy will only be 0.00000001. Four years after that, it will effectively become zero. This will not happen this century, though.</p>

<p>This doesn't mean that there won't be any blocks produced anymore. Assuming Bitcoin is still alive, we'll still have blocks being produced every 10 minutes. Miners will just get all their income from fees, without any subsidy.</p>
 516: <p>Assuming Bitcoin is still active at that point in time, mining will continue, because <a href="https://bitcoin.stackexchange.com/a/112/516">transaction fees will make it worthwhile to do so</a>.</p>

<p>This topic has been discussed heavily in other answers, including:</p>

<ul>
<li><a href="https://bitcoin.stackexchange.com/questions/4796/what-happens-once-mining-reward-gets-cut-in-half">What happens once the mining reward gets cut in half?</a></li>
<li><a href="https://bitcoin.stackexchange.com/questions/161/how-many-bitcoins-will-there-eventually-be">How many bitcoins will there eventually be?</a></li>
<li><a href="https://bitcoin.stackexchange.com/questions/876/how-much-will-transaction-fees-eventually-be">How much will transaction fees eventually be?</a></li>
</ul>

<p>The last link has suggestions that the fee mechanism of Bitcoin may need changing, though that's yet to be proven whether or not it will be a real problem.</p>
