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This article from ArsTechnica is claiming that Bitcoin is not such an environmental disaster, and one of the reasons they mention is that mining will come to an end.

Ok, fair enough, there is an end in mining because the number of bitcoins is fixed but, is there an end to the computation algorithms taking place when generating blocks? My guess was that proof-of-work would still be needed at that time, therefore we would still need ASICs, GPUs or whatnot?

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No, mining will not come to an end at that point. The article is incorrect.

Mining will continue once the block rewards are no longer available, as transaction fees will continue to be offered. As the block rewards tend towards zero, the total value of transaction fees in each block will start to exceed the block reward. This will happen long before the block reward reaches zero if Bitcoin continues to be grow at the current rate.

Mining is required to provide the confirmation that each transaction is valid. If mining was to end, it would be because nobody was submitting transactions any more. This isn't anticipated any time soon!

This chart shows the value of transaction fees per day, in BTC. Currently, about 50BTC per day is paid in fees, which works out to be about 0.34BTC per block (50 divided by 144, which is the number of blocks per day).

If Bitcoin remains at it's currently level of popularity, transaction fees will be larger than the block reward by the year 2040. By the year 2064, more than 99% of miner's rewards would come from transaction fees. After 2140, there will be no more block rewards given.

Related questions:

  • Even if nobody transmits transactions, mining continues. For a significant part of Bitcoin's history, most blocks had no transactions. – Pieter Wuille Apr 16 '13 at 20:27
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    if mining will not come to an end, why is this question duplicate of one that is called "The end of mining"? Sounds like we should edit that question to be "Will mining end?" instead. – knocte Apr 17 '13 at 10:50
  • @PieterWuille and I assume that mining continued because block rewards were high enough to justify mining. But when block rewards have disappeared, mining will only continue if transfer fees justify the mining costs. Or, said otherwise: no block rewards and no transaction fees means no mining. Your "blocks with no transactions" of the "significant part of bitcoin history" had high block rewards. – dangonfast Nov 14 '17 at 10:49
  • @dangonfast Good point. Even though the value of the rewards at some points was low due to a low market price, the design of Bitcoin meant that it would rise if it continued to succeed. In future though, if fees are too low then even if some miners still operate out of the goodness of their hearts it wouldn't be enough to keep the network secure and reliable. – Highly Irregular Nov 14 '17 at 18:07
  • @HighlyIrregular but fees are subject to a regular pricing system offer/demand. If mantaining the blockchain has associated costs, few if any miners will operate with fees below those costs, reducing the amount of miners and thus raising tranasction prices. If the bitcoin system does not tollerate those prices (meaning users do not want to pay the fees) the system will collapse. That said, I think transfer fees will anyway remain at a level corresponding to the sum of current transfer fees + block reward, since that seems to be what makes mining profitable. – dangonfast Nov 14 '17 at 19:24
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Well, there will not be the end of mining, people will continue mining once the block rewards are no longer available, but the transaction fee will be continued to offer. Even though the value of the rewards at some points was low due to a low market price, the design of Bitcoin meant that it would rise if it continued to succeed.

Mining is the process to verify the transaction and check if it is valid. Mining will end only if people will stop doing transactions which are not going to happen anytime soon. In future though, if fees are too low then even if some miners still operate out of the goodness of their hearts it wouldn't be enough to keep the network secure and reliable.

The transaction fee is getting low, and people are again investing in Bitcoin. So I don’t think that in near-future Bitcoin will lose its place. And people will keep on mining with higher profits.

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