Millennium Prize Problems
The Millennium Prize Problems are seven unsolved mathematical problems formally designated by the Clay Mathematics Institute in 2000. Each problem carries a prize of one million US dollars for a correct solution, making them among the most prominent open questions in mathematics. The Institute established this list to highlight fundamental problems that have resisted resolution despite centuries of investigation by mathematicians worldwide, spanning fields including analysis, algebra, geometry, and theoretical computer science.
The Seven Problems
The problems are: the Riemann Hypothesis (concerning the distribution of prime numbers), P versus NP (a central question in computational complexity), the Navier–Stokes existence and smoothness problem (fundamental to fluid dynamics), the Birch and Swinnerton-Dyer conjecture (relating elliptic curves to number theory), Yang–Mills existence and mass gap (from theoretical physics), the Hodge conjecture (algebraic geometry), and the Kato conjecture (also called the Kadison-Singer problem, resolved in 2013). The selection reflects areas where progress would have significant implications across mathematics and science.
Status and Impact
As of the early 2020s, only one problem has been formally solved: the Poincaré conjecture, which was removed from the list after Grigori Perelman’s proof in 2003 (though Perelman declined the prize). The Millennium Problems have influenced mathematical research priorities and funding, despite the monetary incentive being secondary to the intellectual prestige associated with solving them. These problems remain central to contemporary mathematical inquiry and represent genuine barriers to deeper understanding in their respective domains.