Seven Million Dollar Millennium Problems

In order to celebrate mathematics in the new millennium, The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) has named seven Prize Problems. The Scientific Advisory Board of CMI selected these problems, focusing on important classic questions that have resisted solution over the years. The Board of Directors of CMI designated a $7 million prize fund for the solution to these problems, with $1 million allocated to each. During the Millennium Meeting held on May 24, 2000 at the College de France, Timothy Gowers presented a lecture entitled The Importance of Mathematics, aimed for the general public, while John Tate and Michael Atiyah spoke on the problems. The CMI invited specialists to formulate each problem.

One hundred years earlier, on August 8, 1900, David Hilbert delivered his famous lecture about open mathematical problems at the second International Congress of Mathematicians in Paris. This influenced our decision to announce the millennium problems as the central theme of a Paris meeting.

The rules for the award of the prize have the endorsement of the CMI Scientific Advisory Board and the approval of the Directors. The members of these boards have the responsibility to preserve the nature, the integrity, and the spirit of this prize.

Paris, May 24, 2000

Please send inquiries regarding the Millennium Prize Problems to prize.problems@claymath.org.
Birch and Swinnerton-Dyer Conjecture
Hodge Conjecture
Navier-Stokes Equations
P vs NP
Poincare Conjecture
Riemann Hypothesis
Yang-Mills Theory

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Rules
Millennium Meeting Videos

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Riemann Hypothesis
Formulated in his 1859 paper, the Riemann hypothesis in effect says that the primes are distributed as regularly as possible given their seemingly random occurrence on the number line. Riemann's work gave an 'explicit' formula for the number of primes less than x in terms of the zeros of the zeta function. The first term is x/log(x). The Riemann hypothesis is equivalent to the assertion that other terms are bounded by a constant times log(x) times the square root of x. The Riemann hypothesis asserts that all the 'non-obvious' zeros of the zeta function are complex numbers with real part 1/2.