Imagine: With every new Block, new scientific progress is made. I believe this is possible with technology available yet and some not to hard work. Since the beginning of Bitcoin there was the Quest for an Proof of Work system not only useful to Bitcoins security, but other realms too. I propose to use research in Mathematics as replacement or addition for the currently very limited Proof of Work Systems in Cryptocurrencies.

Proof of Work: to find a solution to a problem must be hard to find but easy to verify. In Bitcoins its of the Form: Find a special Value x for that Hash(x) < k holds.

To find a fitting hash is hard, but its easy to proof a found input is correct. The Workinput is allways given by transforming the Last POW Output to the Input of the next Run. Finaly the space requirements are minimal. One Large Input and Outputnumber, and some variable X. Around 80 Bytes. But the Found hashes have no other Value to the World. At some point, further development of ASICs may be fostered by the quest of ever faster hashing. But new motivation isn't really necessary in this field of science. So a lot of energy is used up rather useless beside maybe warming some glasshouses to grow vegetables. One big positiv site of this PoW-system must be named. Botnets have no chance, since highly specialized hardware rather than software is used.

It is not clear, weather for each difficulty an solution can be found!

Essentially everything that is possible to know in mathematics can be derived from a handful of axioms known as Zermelo-Fraenkel set theory, which is the culmination of many years of effort to isolate the essential nature of mathematics and is one of the most profound achievements of mankind[1]. The Project arround Methamath tries to Proof all Math without gaps. A quite big Database off allready proofen Theorems exists, that could be optimized. This would be MathPOW-Optimization. Adding new Proofs to the Database would be MathPOW-Research.

Essentialy the search for the proof of an Theorem builds a Searchtree. Every Node of the Tree is the result of the Application of a replacement rule. At the Moment, Humans are better at Finding the Right Path on this Searchtree, but in Principle its possible for Computers too. Possible Mathrelated tasks as Proof of Work are:

• Searching Proofs for Mathematical Theorems. They are hard to find, but once found, easy to verify. (If written in a computerreadable form.)
• Searching shorter versions of known Proofs.
• Finding Theorems, that can be used to shorten multible know Proofs. Such Theorems may be a specialy interesting.

• MathPOW is generally usefull.
• MathPOW is workable by humans.
• MathPOW fosters research in automatic Theoremproofing and mathematical education.

• Not possible to adjust the difficulty to adjuste the rate of found solutions
  • Combination of ordenary POW and MathPOW.
  • IF rate to high, demand multiple new Proofs, and if rate to low, increase reward.
• How to generate new Theorems?
  • Not solvable by the Miners, they could collide to create easy Problmes.
• When a shorter Proof is found, it could be artifical made longer, to resubmit it later a little bit shorter again...
  • Reward is a Function of the lengt of a Mathematical Proof. Only the diffrenece between old and new value is payed.
    • Could still be abused to create manny Blocks for the Transactionfees.
• When the Blockchain forks, with normal POW you have to decide, on wich Fork you work, with MathPOW thats not the case, proof would be valid on both forks. Thats the same weakness like with Proof of Stake
  • Combination of PoW and MathPoW save?

• Generalizations of multiple theorems into one as MathPoW?

This could be implemented into Bitcoin, its the most Valueable chain, and so would create most gain for research. As Sidechain, or in the Mainchain as an Additional kind of POW. I belive this would increase the respekt for Bitcoin, and therefor the Marketcap.

The transition must be partial, the gain in recocnition of Bitcoin and so Value has to be bigger as the loss of current Miners, a slow transiton is the only way for Bitcoin to stay save. Maybe of 0.5% per Month

• tell all your Cryptocurrenydevelopers that you want MathPOW!
• Create Pullrequest with new Ideas or corrected Spelling ;)
• Help me to work on MathPOW by donating to 1CfD6YX9Z5gjNAetDdmrdxJrHTt6j4NhAr

Not every computeintens operation is usable as Proof of Work. A Worktask has to satisfy the following points:

• Hard to Compute but easy to verify: Although any given solution to such a problem can be verified quickly, there is no known efficient way to locate a solution in the first place. Such Problems are part of the complexity class NP or NP-Hard as long as they are not part of P.

• Abundant Input: You must be able to generate your work yourself. Else you may would have to wait for some third person to provide Work for you. That may not be possible in a trust free manner.

• small memory-space requirements: Because all the Proof of Work Inputs and Results have to be saved in the Blockchain to be able to verify the consistency of it in a trust free manner.

• adjustable worksize: Otherwise the Cryptocurrencynetwork wouldn't be able to set the median time that passes between two found solutions/blocks.

• can be parallelized: Work has to be derivative from a nuance that can be free chosen, so miners can start searching on different parts of the workinputspace, or the problem itself has to offer many points to start from.

• (Can be satisfied) in the sense, that an solution to an Problem must be foundeable in any case. Else it would be possible to get stuck in an search for a solution for an input for ever.

• (No pre-calculating of future Blocks:) It should not be possible, to pre-calculate to reduce the work of a specific future block beside the current one. That does not mean, it should not be possible to make the search for an solution more efficient by calculating some general information usable for every future block.(Hash-tables or so).

• save from the invasion of botnets
• suitable for pooled mining to combine computing power to reduce payment variation
• unsuitable for pooled mining to combat minercentralisation

• Primecoin [http://primecoin.io/] PoW is to find Primenumberpairs.

• Gridcoin [http://www.gridcoin.us/] Not Trustless. Work is recived from BIONC servers. Use of Whitelists.

[1] http://us.metamath.org/mpegif/mmset.html#theorems