The Most Difficult Unsolved Mathematical Problem, Ranked

Choose the unsolved mathematical problem you think is the most difficult!

Author: Gregor Krambs
Updated on Jun 10, 2024 06:34
Ranking the most challenging unsolved mathematical problems serves a distinct purpose in the academic community. It highlights areas where curiosity and expertise can combine to push the boundaries of knowledge. By identifying which problems are deemed the most formidable, both aspiring and seasoned mathematicians can see where their efforts might contribute most significantly. This voting system allows everyone, from novices to experts in the field, to express their views on which mathematical conundrums they believe should be prioritized or are of greatest interest. This collective insight not only enriches the mathematical community but also guides future explorations. It's an opportunity for all to engage with the frontiers of mathematical thought and to potentially influence the direction of future research.

What Is the Most Difficult Unsolved Mathematical Problem?

  1. 1

    Riemann Hypothesis

    Proposes that all non-trivial zeros of the Riemann zeta function have their real parts equal to 1/2.
    • Proposed by: Bernhard Riemann in 1859
    • Field: Number Theory
  2. 2

    Birch and Swinnerton-Dyer Conjecture

    Predicts a way to determine the number of rational points on a given elliptic curve based on the behavior of the curve's L-function at s=1.
    • Field: Number Theory
  3. 3

    Hodge Conjecture

    Concerns the relationship between algebraic cycles and cohomology, specifically predicting which de Rham cohomology classes are algebraic.
    • Field: Algebraic Geometry
  4. 4

    Yang-Mills Existence and Mass Gap

    Involves proving the existence of a quantum field theory for Yang-Mills fields and establishing a non-zero mass gap for the theory.
    • Field: Theoretical Physics
  5. 5

    Goldbach's Conjecture

    Asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers.
    • Proposed by: Christian Goldbach in 1742
    • Field: Number Theory
  6. 6

    Twin Prime Conjecture

    Suggests that there are infinitely many prime pairs that have a difference of 2.
    • Field: Number Theory
  7. 7

    Poincaré Conjecture

    Proposed that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. It has been proven, thus no longer unsolved.
    • Proposed by: Henri Poincaré in 1904
    • Solved by: Grigori Perelman in 2003
    • Field: Topology
  8. 8

    Collatz Conjecture

    Posits that the Collatz sequence for any positive integer will eventually reach the number 1.
    • Proposed by: Lothar Collatz in 1937
    • Field: Number Theory
  9. 9
  10. 10

    Navier-Stokes Existence and Smoothness

    Concerns the existence and smoothness of solutions to the Navier-Stokes equations, which describe the motion of fluid substances.
    • Field: Fluid Dynamics

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About this ranking

This is a community-based ranking of the most difficult unsolved mathematical problem. We do our best to provide fair voting, but it is not intended to be exhaustive. So if you notice something or puzzle is missing, feel free to help improve the ranking!


  • 162 votes
  • 10 ranked items

Voting Rules

A participant may cast an up or down vote for each puzzle once every 24 hours. The rank of each puzzle is then calculated from the weighted sum of all up and down votes.

Additional Information

More about the Most Difficult Unsolved Mathematical Problem

Riemann Hypothesis
Rank #1 for the most difficult unsolved mathematical problem: Riemann Hypothesis (Source)
Mathematics has always been a field of great challenges. Among these challenges, some problems remain unsolved. These problems intrigue mathematicians and spark their curiosity. They push the boundaries of human knowledge and understanding.

These problems often have simple statements. Yet, their solutions elude even the brightest minds. They involve basic concepts but require deep insights. The search for solutions drives much of mathematical research. It leads to new methods and theories.

One reason these problems are hard is their abstract nature. They often deal with infinite sets or complex structures. Understanding these requires advanced tools. Developing these tools can take years or even decades.

Another reason is the interconnectedness of mathematics. Solving one problem can depend on progress in many areas. This requires a broad knowledge base. Mathematicians must collaborate and share ideas. This collective effort is crucial.

These unsolved problems also have practical implications. They can impact fields like cryptography, physics, and computer science. Solving them can lead to technological advances. This adds to their importance and urgency.

Despite the challenges, progress is made. Mathematicians develop new techniques and approaches. They refine existing theories and uncover new connections. This slow, steady progress is part of the beauty of mathematics.

Students and researchers are drawn to these problems. They represent the frontier of human knowledge. Solving one can bring great recognition. It can also bring a sense of accomplishment and satisfaction.

The history of mathematics shows that no problem is unsolvable. Many problems once thought impossible have been solved. This gives hope and motivation to those working on today’s unsolved problems. They know that persistence and creativity can lead to breakthroughs.

In the end, the pursuit of these problems is about more than finding solutions. It is about the journey of discovery. It is about pushing the limits of what we know. It is about the joy of exploring the unknown. This is what makes mathematics a living, evolving field.

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