Unsolved mathematical equations with rewards for solving
Executive summary
The most prominent set of unsolved mathematical problems that carry monetary rewards are the seven Millennium Prize Problems, each carrying a US$1,000,000 award for the first correct solution as established by the Clay Mathematics Institute in 2000, for a total $7 million fund [1][2]. One of the seven—the Poincaré Conjecture—has been solved; Grigori Perelman’s proof was accepted but he declined the Clay prize, leaving six active million-dollar puzzles [3][4][5].
1. The prize and its pedigree: an institutional bet on hard problems
The Clay Mathematics Institute selected seven “millennium” problems to spotlight enduring frontiers of mathematics and allocated $1 million to the solver of each problem as a public incentive to advance the field, following a long history of problem-based prizes in mathematics [1][2][6].
2. What remains unsolved: the six active Millennial enigmas
Six of the original seven problems remain officially unsolved: the Riemann Hypothesis, P versus NP, Navier–Stokes existence and smoothness, Birch and Swinnerton‑Dyer conjecture, the Hodge Conjecture, and the Yang–Mills existence and mass gap—each described in the Clay/AMS statements and popular summaries as deep, widely consequential questions in number theory, computer science, analysis, algebraic geometry, and mathematical physics [3][2][7].
3. Short guide to the mathematical stakes of each problem
P versus NP asks whether every problem whose solution can be verified quickly by a computer can also be solved quickly, a question with profound implications for cryptography and algorithms [7][8]; the Riemann Hypothesis concerns the distribution of prime numbers and underlies much analytic number theory [2][8]; Navier–Stokes asks whether smooth solutions always exist for the fundamental equations of fluid dynamics in three dimensions [9][8]; Birch and Swinnerton‑Dyer predicts a precise relationship between rational solutions on elliptic curves and the behavior of an associated L‑function [3][8]; the Hodge Conjecture links topology and algebraic geometry through the nature of certain cycles on algebraic varieties [2]; Yang–Mills existence and mass gap asks for a rigorous quantum field theory with a positive mass gap matching particle physics intuition [9][2].
4. Money, motivation and controversy: does a prize help or distort?
The million‑dollar incentive has raised public awareness and encouraged work on these questions, but voices within mathematics have warned that attaching a monetary prize risks appropriating collective research agendas and could subtly shift incentives away from curiosity‑driven exploration—critiques recorded in commentary about the Clay Institute’s initiative [4].
5. The Perelman precedent: proof accepted, prize refused
Grigori Perelman produced a proof of the Poincaré Conjecture that completed Hamilton’s program using Ricci flow; the mathematical community accepted the proof and the Clay Institute awarded the Millennium Prize for that problem, but Perelman declined both the Fields Medal earlier and the Clay prize, citing philosophical objections [3][4][5].
6. Beyond the seven: prizes, history and the culture of problem‑solving
Prize problems have a long pedigree in mathematics—past competitions yielded major discoveries—and the Millennium Problems were explicitly modeled to rekindle public interest and signal that deep, open questions remain central to the field, even as new unsolved problems continue to emerge outside the Clay list [6][7].
7. Practical realities for would‑be solvers
Solving one of these problems typically requires deep, sustained research, community verification, and publication meeting the Clay Institute’s formal criteria; the prize is awarded only once a solution attains broad acceptance in the mathematical literature [1][10]. Exact award rules and problem statements are maintained by the Institute and associated mathematical bodies [1][7].