Millennium Prize Problems

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The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of August 2010, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. Only the Poincaré conjecture has been solved, by Grigori Perelman, who declined the award.

The seven problems are:

Contents

P versus NP

The question is whether, for all problems for which a computer can verify a given solution quickly (that is, in polynomial time), it can also find that solution quickly. The former describes the class of problems termed NP, whilst the latter describes P. The question is whether or not all problems in NP are also in P. This is generally considered the most important open question in mathematics and theoretical computer science as it has far-reaching consequences in mathematics, biology, philosophy[citation needed] and cryptography (see P versus NP problem proof consequences).

If the question of whether P=NP were to be answered affirmatively it would trivialise the rest of the Millenium Prize Problems (and indeed all but the unprovable propositions in mathematics) because they would all have direct solutions easily solvable by a formal system.

Mathematicians and Computer Scientists expect that the statement 'P=NP' will be shown to be false.

The official statement of the problem was given by Stephen Cook.

The Hodge conjecture

The Hodge conjecture is that for projective algebraic varieties, Hodge cycles are rational linear combinations of algebraic cycles.

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