Where to Find Open Problems in Mathematics

Why Open Problems Matter

Mathematics is not a finished subject. There are deep, important questions in every area of mathematics that remain unanswered — some stated centuries ago, others discovered just recently. Knowing about these open problems is valuable for several reasons:

• Motivation. Seeing what is unknown gives you a sense of where mathematics is going and why the tools you are learning matter.
• Research direction. If you are considering graduate school, understanding the open questions in different areas helps you choose a field.
• Inspiration. Some open problems are simple to state but incredibly deep. Thinking about them, even without solving them, develops your mathematical thinking.

This guide organizes the best resources for finding open problems, from famous prize problems to accessible collections for undergraduates.

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The Great Prize Problems

The Millennium Prize Problems

The Clay Mathematics Institute Millennium Prize Problems

In the year 2000, the Clay Mathematics Institute (CMI) identified seven problems, each carrying a $1,000,000 prize for a correct solution. As of 2026, six remain unsolved:

1. P vs NP — Is every problem whose solution can be verified quickly also solvable quickly?
2. The Riemann Hypothesis — Do all non-trivial zeros of the Riemann zeta function lie on the critical line $\text{Re}(s) = \frac{1}{2}$?
3. Yang-Mills Existence and Mass Gap — Prove that the quantum Yang-Mills theory exists and has a positive mass gap.
4. Navier-Stokes Existence and Smoothness — Do smooth solutions to the Navier-Stokes equations always exist in three dimensions?
5. Birch and Swinnerton-Dyer Conjecture — Is the rank of an elliptic curve determined by the behavior of its $L$-function at $s = 1$?
6. Hodge Conjecture — Are certain cohomology classes on algebraic varieties always representable by algebraic cycles?

The seventh problem, the Poincaré Conjecture, was solved by Grigori Perelman in 2003.

Hilbert's Problems

In 1900, David Hilbert presented 23 problems that he believed would shape the future of mathematics. Many have been solved, some remain open, and several led to entirely new branches of mathematics. A study of Hilbert's problems is a tour through 20th-century mathematics.

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Open Problem Collections by Area

Number Theory

Number theory has some of the most famous and accessible open problems:

• Goldbach's Conjecture: Every even integer greater than 2 is the sum of two primes. Stated in 1742, still unproven.
• Twin Prime Conjecture: There are infinitely many pairs of primes that differ by 2 (like 11 and 13, or 29 and 31).
• Collatz Conjecture: Start with any positive integer. If even, divide by 2. If odd, multiply by 3 and add 1. Does this process always eventually reach 1?
• Legendre's Conjecture: Is there always a prime between $n^2$ and $(n+1)^2$ for every positive integer $n$?

The Open Problem Garden maintains a searchable database of open problems including many in number theory.

Combinatorics

• Hadwiger's Conjecture: Every $k$-chromatic graph contains a $K_k$ minor.
• The Union-Closed Sets Conjecture (Frankl's Conjecture): For any finite family of sets closed under union, some element belongs to at least half the sets.
• The Erdős-Faber-Lovász Conjecture: (Recently settled for large $n$.) Any $n$ complete graphs on $n$ vertices, pairwise sharing at most one vertex, can be properly edge-colored with $n$ colors.

Analysis

• The Kakeya Conjecture: In $\mathbb{R}^n$ for $n \ge 3$, does every Besicovitch set (containing a unit line segment in every direction) have Hausdorff dimension $n$?
• Carleson's Problem (on the sphere): Various extensions of Carleson's theorem on pointwise convergence of Fourier series remain open.
• The Invariant Subspace Problem: Does every bounded linear operator on a separable Hilbert space have a non-trivial closed invariant subspace?

Topology and Geometry

• The Smooth Poincaré Conjecture in Dimension 4: Is every closed, simply connected, smooth 4-manifold with the homology of $S^4$ diffeomorphic to $S^4$?
• The Borel Conjecture: Are aspherical closed manifolds determined up to homeomorphism by their fundamental groups?
• The Volume Conjecture: A conjecture relating the colored Jones polynomial of a knot to the hyperbolic volume of its complement.

Algebra

• The Jacobian Conjecture: If a polynomial map $F : \mathbb{C}^n \to \mathbb{C}^n$ has a nonzero constant Jacobian determinant, is $F$ invertible?
• Kaplansky's Conjectures: Several conjectures about group rings, some of which remain open.

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Where to Find Open Problems Online

The Open Problem Garden

openproblemgarden.org

A wiki-style database of open problems in mathematics, organized by topic. Each problem includes a statement, context, and references.

MathOverflow

mathoverflow.net

The "open-problem" tag on MathOverflow contains hundreds of research-level open questions, often with expert commentary. This is one of the best places to discover problems that are actively being worked on.

Search for [open-problem] on MathOverflow to browse.

arXiv Survey Papers

Many survey papers on arXiv include sections listing open problems in their area. Searching arXiv for "open problems in [area]" often yields excellent results. For example:

• "Open problems in algebraic topology"
• "Some open problems in combinatorics"
• "Open questions in number theory"

Richard Guy's Unsolved Problems in Number Theory

Richard Guy's book Unsolved Problems in Number Theory is a classic collection of open problems, many accessible to undergraduates. While the book itself is not free, summaries of many problems appear in the OEIS and on Wikipedia.

The Kourovka Notebook

For group theory, the Kourovka Notebook is a long-standing collection of open problems updated every few years. It is the definitive source for unsolved problems in group theory and related areas.

AIM Problem Lists

The American Institute of Mathematics hosts problem lists arising from workshops on specific topics. These are curated by experts and represent active research frontiers.

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Open Problems Accessible to Undergraduates

Not all open problems require years of graduate study. Here are some that can be understood with an undergraduate background:

The Collatz Conjecture

Take any positive integer $n$. If $n$ is even, replace it with $n/2$. If $n$ is odd, replace it with $3n + 1$. Repeat. The conjecture states that you always eventually reach 1.

Despite its elementary statement, this problem has resisted all attempts at proof. Paul Erdős said, "Mathematics may not be ready for such problems."

The Lonely Runner Conjecture

Suppose $k$ runners start at the same point on a circular track of unit circumference, each running at a different constant speed. The conjecture states that for each runner, there exists a time when that runner is at distance at least $\frac{1}{k}$ from all other runners.

This is proven for $k \le 7$ but remains open in general.

The Moving Sofa Problem

What is the largest area of a rigid planar shape that can be moved around a right-angled corner in a hallway of width 1? The best known shape (Gerver's sofa) has area approximately 2.2195, but it is not known whether this is optimal.

Happy Ending Problem Generalizations

Erdős and Szekeres proved that for any integer $n \ge 3$, any sufficiently large set of points in general position contains $n$ points forming a convex polygon. The tight bound is known for $n = 3, 4, 5$ and recently for $n = 6$ but remains open for larger $n$.

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How to Approach Open Problems

1. Understand the Problem Thoroughly

Before attempting any ideas, make sure you understand:

• The precise statement of the problem
• Why it is considered important
• What partial results are known
• What approaches have been tried and why they failed

2. Study the Background

Read survey papers and relevant textbook chapters. Understand the tools that have been used on similar problems.

3. Experiment

Use computational tools like SageMath, Mathematica, or Python to explore examples and test conjectures. Many breakthroughs start with computational experimentation.

4. Start with Simpler Versions

Can you solve a special case? A lower-dimensional version? A weaker statement? Building up from simpler cases often reveals the key ideas.

5. Talk to Others

Discuss the problem with peers, attend seminars, and eventually talk to experts. Mathematics is a collaborative endeavor.

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Final Thoughts

Open problems are the lifeblood of mathematics. They drive research, inspire new theories, and connect different areas of the subject. You do not need to be a Fields Medalist to engage with them. Start by reading about the famous problems, explore the accessible ones, and let your curiosity guide you deeper into the mathematics.

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References

• Clay Mathematics Institute — Millennium Prize Problems
• Hilbert's Problems
• Open Problem Garden
• MathOverflow — Open Problems Tag
• Kourovka Notebook — Unsolved Problems in Group Theory
• American Institute of Mathematics — Problem Lists
• OEIS — On-Line Encyclopedia of Integer Sequences
• Richard Guy, Unsolved Problems in Number Theory (Springer)
• arXiv — Mathematics
• Wikipedia — List of Unsolved Problems in Mathematics