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Unsolved Problems in Mathematics

The problems below have resisted rigorous proofs despite significant progress. Each description includes a brief summary and a reference for further study.

1. Riemann Hypothesis

The hypothesis asserts that every nontrivial zero of the Riemann zeta function lies on the critical line with real part one-half. Resolving it would illuminate the distribution of prime numbers and has profound consequences throughout analytic number theory. 1

2. P vs NP

This problem asks whether every problem whose solution can be verified quickly (in polynomial time) can also be solved quickly. A proof either way would reshape computer science, impacting cryptography, optimisation and complexity theory at large. 2

3. Birch and Swinnerton-Dyer Conjecture

For an elliptic curve over the rationals, the conjecture links the number of independent rational points to the behaviour of the curve’s L-function at s = 1. It provides a deep connection between arithmetic geometry and analytic objects. 3

4. Hodge Conjecture

The conjecture proposes that certain cohomology classes on a non-singular projective variety can be represented by algebraic cycles. It remains one of the central open problems in algebraic geometry. 4

5. Yang–Mills Existence and Mass Gap

This problem seeks a rigorous quantum field theory for non-abelian gauge fields that exhibits a mass gap, meaning particles possess positive lower bounds for their masses. It is crucial for understanding the mathematical foundations of particle physics. 5

6. Navier–Stokes Existence and Smoothness

The question asks whether smooth solutions to the three-dimensional Navier–Stokes equations exist for all time, or if singularities can form from smooth initial conditions. A resolution would clarify the mathematics of fluid flow. 6

7. Goldbach’s Conjecture

Goldbach proposed that every even integer greater than two is the sum of two primes. Despite overwhelming numerical evidence and partial results, a general proof remains unknown. 7

8. Twin Prime Conjecture

The conjecture posits that there exist infinitely many pairs of primes separated by two. Recent advances have narrowed the permissible gap, but infinitude is still unproven. 8

9. Collatz Conjecture

Starting from any positive integer, repeatedly applying the Collatz map (halve even numbers, triple odd numbers and add one) appears to eventually reach one. Proving this behaviour for every starting value has eluded mathematicians for decades. 9

10. abc Conjecture

This conjecture links the prime factors of three integers a, b and c satisfying a + b = c. It predicts that, except for finitely many triples, c cannot be too large relative to the product of the distinct prime factors of abc. Its resolution would unify numerous results in number theory. 10

Footnotes

  1. Clay Mathematics Institute, The Riemann Hypothesis (Millennium Prize Problem description). ↩

  2. Clay Mathematics Institute, P vs NP Problem (Millennium Prize Problem description). ↩

  3. Clay Mathematics Institute, The Birch and Swinnerton-Dyer Conjecture (Millennium Prize Problem description). ↩

  4. Clay Mathematics Institute, The Hodge Conjecture (Millennium Prize Problem description). ↩

  5. Clay Mathematics Institute, Yang–Mills and Mass Gap (Millennium Prize Problem description). ↩

  6. Clay Mathematics Institute, Navier–Stokes Equation (Millennium Prize Problem description). ↩

  7. Helmut Koch, Number Theory: Algebraic Numbers and Functions, Chapter 7. ↩

  8. James Maynard, “Small gaps between primes,” Annals of Mathematics 181 (2015). ↩

  9. Jeffrey C. Lagarias, “The 3x + 1 Problem: An Annotated Bibliography (1963–1999),” arXiv:math/0309224. ↩

  10. David Masser and Joseph Oesterlé, conference reports (1985) introducing the abc conjecture. ↩