In 1955, mathematician John Nash wrote a letter to the NSA, where he speculated that cracking a sufficiently complex code would require time exponential in the length of the key.
Another mention of the underlying problem occurred in a 1956 letter written by Kurt Gödel to John von Neumann.
Fischer and Rabin proved in 1974 that every algorithm that decides the truth of Presburger statements of length n has a runtime of at least 2^{2^{cn}} for some constant c.
Many of these problems are #P-complete, and hence among the hardest problems in #P, since a polynomial time solution to any of them would allow a polynomial time solution to all other #P problems. ==Problems in NP not known to be in P or NP-complete== In 1975, Richard E.
Arguably, the biggest open question in theoretical computer science concerns the relationship between those two classes: Is P equal to NP? Since 2002, William Gasarch has conducted three polls of researchers concerning this and related questions.
Confidence that P ≠ NP has been increasing – in 2019, 88% believed P ≠ NP, as opposed to 83% in 2012 and 61% in 2002.
For example, in 2002 these statements were made: ==Consequences of solution== One of the reasons the problem attracts so much attention is the consequences of the some possible answers.
A Princeton University workshop in 2009 studied the status of the five worlds. ==Results about difficulty of proof== Although the P = NP problem itself remains open despite a million-dollar prize and a huge amount of dedicated research, efforts to solve the problem have led to several new techniques.
Confidence that P ≠ NP has been increasing – in 2019, 88% believed P ≠ NP, as opposed to 83% in 2012 and 61% in 2002.
Woeginger maintains a list that, as of 2018, contains 62 purported proofs of P = NP, 50 proofs of P ≠ NP, 2 proofs the problem is unprovable, and one proof that it is undecidable.
Confidence that P ≠ NP has been increasing – in 2019, 88% believed P ≠ NP, as opposed to 83% in 2012 and 61% in 2002.
When restricted to experts, the 2019 answers became 99% believe P ≠ NP.
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