The Most Difficult Math Equation, Ranked

Choose the math equation you think is the most difficult!

Author: Gregor Krambs
Updated on Jun 20, 2024 06:31
Mathematics often presents challenges that push the boundaries of understanding and problem-solving. Among these, certain equations stand out due to their complexity and the extensive cognitive effort required to solve them. By ranking these formidable math problems, individuals can appreciate the hierarchy of difficulty and the intellectual milestones in the field of mathematics. This interactive listing allows users to engage by voting for what they consider the most challenging equations. This continuous input helps create a dynamic and updated ranking that reflects the collective opinion of a diverse audience. As more enthusiasts participate, the accuracy of the rankings improves, providing a clearer picture of the mathematical landscape's most demanding challenges.

What Is the Most Difficult Math Equation?

  1. 1

    Riemann Hypothesis

    Suggests that all non-trivial zeros of the Riemann zeta function have their real parts equal to 1/2.
    • Field: Number theory
    • Millennium Prize Problem: Yes
  2. 2

    Goldbach's Conjecture

    Asserts that every even integer greater than 2 can be expressed as the sum of two prime numbers.
    • Field: Number theory
    • Stated: 1742
  3. 3

    Four Color Theorem

    States that any map in a plane can be colored using four colors in such a way that regions sharing a common boundary (other than a single point) do not share the same color.
    • Field: Graph theory
    • Proven: 1976
  4. 4

    Hilbert's Sixteenth Problem

    Asks for the maximum number and relative positions of certain types of limit cycles on a plane polynomial vector field.
    • Field: Differential equations
    • Part of Hilbert's Problems: Yes
  5. 5

    Navier-Stokes Existence and Smoothness

    Concerns the existence and smoothness of solutions to the Navier-Stokes equations, which describe the motion of viscous fluid substances.
    • Field: Fluid dynamics
    • Millennium Prize Problem: Yes
  6. 6
  7. 7

    Hodge Conjecture

    Predicts that for certain classes of non-singular projective algebraic varieties, the Hodge cycles are rational linear combinations of algebraic cycles.
    • Field: Algebraic geometry
    • Millennium Prize Problem: Yes
  8. 8

    Collatz Conjecture

    Involves a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1.
    • Field: Number theory
    • Simplicity: Deceptively simple
  9. 9

    Yang-Mills Existence and Mass Gap

    Involves proving the existence of a mass gap in the quantum field theory of Yang-Mills fields.
    • Field: Theoretical physics
    • Millennium Prize Problem: Yes
  10. 10

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About this ranking

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


  • 32 votes
  • 10 ranked items

Voting Rules

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

Additional Information

More about the Most Difficult Math Equation

Riemann Hypothesis
Rank #1 for the most difficult math equation: Riemann Hypothesis (Source)
Mathematics is a field full of challenges. Some equations are so complex that they baffle even the sharpest minds. These equations often arise in areas like physics, engineering, and economics. They can describe natural phenomena, predict outcomes, or solve practical problems. The hardest equations often involve many variables and intricate relationships.

One reason these equations are difficult is their abstract nature. They may not have a clear physical interpretation. This makes it hard to visualize or understand their behavior. Mathematicians must rely on pure logic and creativity to tackle them. Another reason is the sheer size and complexity of the calculations involved. Some equations require advanced computational methods to solve. These methods may involve breaking the problem into smaller, more manageable parts.

Many of these challenging equations have a long history. They often build on the work of previous mathematicians. Over time, new techniques and tools have emerged to address them. Despite this progress, some equations remain unsolved for decades or even centuries. This ongoing struggle highlights the depth and richness of mathematics as a discipline.

Solving these equations can have significant implications. In physics, for example, a solution might lead to new insights about the universe. In engineering, it could result in more efficient designs or processes. In economics, it might help predict market trends or optimize resource allocation. The potential benefits drive mathematicians to keep pushing the boundaries of their field.

Collaboration plays a key role in tackling difficult equations. Mathematicians often work in teams, sharing ideas and approaches. They may also draw on insights from other disciplines. This interdisciplinary approach can lead to breakthroughs. Conferences and publications provide platforms for sharing progress and challenges.

Despite the difficulties, mathematicians find beauty in these equations. They appreciate the elegance and symmetry that often emerge. This aesthetic appeal can be a powerful motivator. It drives them to keep searching for solutions, even when progress is slow.

Education and training are crucial for preparing mathematicians to tackle these problems. Advanced courses and research opportunities help students develop the necessary skills. Mentorship from experienced mathematicians also plays a vital role. This guidance can provide valuable insights and encouragement.

The journey to solve a difficult equation can be long and arduous. It requires patience, persistence, and a deep love for the subject. Mathematicians must be willing to explore many different avenues. They must also be prepared to face setbacks and dead ends. Each step, however small, brings them closer to a solution.

In conclusion, the most difficult math equations present formidable challenges. They require a blend of logic, creativity, and collaboration. Solving them can lead to significant advancements in various fields. The journey is demanding, but the rewards are immense. Mathematicians continue to strive for solutions, driven by curiosity and a passion for discovery.

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