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Proof of work (PoW) is a form of cryptographic proof in which one party (the prover) proves to others (the verifiers) that a certain amount of a specific computational effort has been expended. [1] Verifiers can subsequently confirm this expenditure with minimal effort on their part.
In proof by exhaustion, the conclusion is established by dividing it into a finite number of cases and proving each one separately. The number of cases sometimes can become very large. For example, the first proof of the four color theorem was a proof by exhaustion with 1,936 cases. This proof was controversial because the majority of the cases ...
Langlands's proof of the functional equation for Eisenstein series was 337 pages long. 1983 Trichotomy theorem. Gorenstein and Lyons's proof for the case of rank at least 4 was 731 pages long, and Aschbacher's proof of the rank 3 case adds another 159 pages, for a total of 890 pages. 1983 Selberg trace formula. Hejhal's proof of a general form ...
In this example, although S(k) also holds for {,,,,}, the above proof cannot be modified to replace the minimum amount of 12 dollar to any lower value m. For m = 11 , the base case is actually false; for m = 10 , the second case in the induction step (replacing three 5- by four 4-dollar coins) will not work; let alone for even lower m .
Hashcash is a proof-of-work system used to limit email spam and denial-of-service attacks. Hashcash was proposed in 1997 by Adam Back [1] and described more formally in Back's 2002 paper "Hashcash – A Denial of Service Counter-Measure". [2] In Hashcash the client has to concatenate a random number with a string several times and hash this new ...
Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Sir Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were believed to be impossible to prove using previous ...
One well-known example of a zero-knowledge proof is the "Where's Waldo" example. In this example, the prover wants to prove to the verifier that they know where Waldo is on a page in a Where's Waldo? book, without revealing his location to the verifier. [9] The prover starts by taking a large black board with a small hole in it, the size of Waldo.
Indeed, the above proof that the law of excluded middle implies proof by contradiction can be repurposed to show that a decidable proposition is ¬¬-stable. A typical example of a decidable proposition is a statement that can be checked by direct computation, such as " n {\displaystyle n} is prime" or " a {\displaystyle a} divides b ...
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