The Most Difficult Equation to Solve, Ranked

Voting rules: Choose the equation you think is the most difficult!

Updated on May 3, 2024 06:24

In the world of mathematics, equations often stand as towering challenges that attract brilliant minds seeking to test their limits. The quest to solve these complex problems drives innovation and deepens our understanding of mathematical principles. However, determining which equation has stumped the greatest number of mathematicians can provide insights into where the edges of human knowledge and capability currently lie. This ranking invites enthusiasts, experts, and curious minds alike to cast their votes on which equations they believe are the most formidable. By participating, users contribute to a collective assessment that highlights these daunting mathematical challenges. Such engagement not only aids in recognizing the intricacies of these equations but also energizes the community to further investigate and potentially solve these significant puzzles.

What Is the Most Difficult Equation to Solve?

1

49
votes
Navier-Stokes equation

This equation describes the motion of fluids and is notoriously difficult to solve due to its non-linear nature. It is one of the Millennium Prize Problems, with a million-dollar prize for its solution.
The Navier-Stokes equation is a set of partial differential equations that describe the motion of viscous fluid substances. It is widely used in fluid dynamics and is considered one of the most important and beautiful equations in mathematics.

Type: Partial differential equations

Domain: Fluid dynamics

Variables: Velocity, pressure, density, viscosity

Equation form: Conservation of momentum and mass

Nonlinearity: Nonlinear equations
Navier-Stokes equation in other rankings
- Position 4 of 10 in Most beautiful equation in mathematics
2

53
votes
Hilbert's tenth problem

David Hilbert

This problem asks whether there exists a general algorithm for determining whether a given Diophantine equation has a solution. It was proved to be unsolvable by Yuri Matiyasevich in 1970.
Hilbert's tenth problem, also known as the Entscheidungsproblem, is a famous mathematical problem posed by David Hilbert in 1900. It asks whether there exists an algorithm that can determine whether a given Diophantine equation has integer solutions or not.

Problem type: Decision problem

Problem domain: Number theory

Equation type: Diophantine equation

Problem statement: Given a Diophantine equation, determine if it has integer solutions

Formalization: Existential quantification over integers
3

21
votes
Fermat's Last Theorem

Pierre de Fermat

This equation states that there are no whole number solutions to the equation x^n + y^n = z^n for n > 2. It was famously proved by Andrew Wiles in 1995 after over 350 years of attempts by mathematicians.
Fermat's Last Theorem is a famous mathematical statement that was first proposed by Pierre de Fermat in 1637. It states that no three positive integers a, b, and c can satisfy the equation a^n + b^n = c^n for any integer value of n greater than 2.

Conjecture: a^n + b^n = c^n has no integer solutions for n > 2

Proof: The proof was completed by Andrew Wiles with assistance from Richard Taylor

Year of Proof: 1994

Field: Number Theory

Significance: One of the most famous and long-standing unsolved problems in mathematics
Fermat's Last Theorem in other rankings
- Position 1 of 10 in Most advanced mathematical proof used in reasoning
- Position 4 of 10 in Most difficult equation
- Position 4 of 10 in Most difficult math equation
- Position 5 of 10 in Most difficult math question
4

16
votes
Riemann Hypothesis

Bernhard Riemann

This conjecture concerns the distribution of prime numbers and is considered one of the most important unsolved problems in mathematics. It has far-reaching implications for cryptography and computer science.
The Riemann Hypothesis is one of the most significant unsolved problems in mathematics. It is a conjecture made by German mathematician Bernhard Riemann in 1859, regarding the distribution of prime numbers. Essentially, it provides information about the behavior of the Riemann zeta function, which is a complex-valued function that holds great importance in number theory. The Riemann Hypothesis suggests that all non-trivial zeros of the zeta function have a real part of 1/2.

Conjecture Type: Unsolved

Field: Number Theory

Conjectured Date: 1859

Importance: Critical

Significance: Determines prime numbers distribution
Riemann Hypothesis in other rankings
- Position 1 of 10 in Most difficult unsolved mathematical problem
- Position 2 of 10 in Most difficult math equation
- Position 2 of 10 in Most difficult equation
- Position 3 of 10 in Most advanced mathematical proof used in reasoning
5

17
votes
Yang-Mills existence and mass gap

Yang, Mills

This problem seeks to prove the existence of a mass gap in quantum Yang-Mills theory, which describes the interactions between elementary particles. It is also one of the Millennium Prize Problems.
Yang-Mills existence and mass gap is a longstanding mathematical problem in quantum field theory. It seeks to understand the existence and properties of solutions to the Yang-Mills equations and the nature of the mass gap in the quantum chromodynamics (QCD) theory of strong interactions.

Equations: Yang-Mills equations

Theory: Quantum field theory

Application: Quantum Chromodynamics (QCD)

Problem: Existence and properties of solutions

Focus: Nature of the mass gap
6

10
votes
P versus NP

Stephen Cook and Leonid Levin

This is a problem in computer science that asks whether certain complex problems that are difficult to solve can be solved efficiently with a computer. It has implications for cryptography, artificial intelligence, and many other fields.
P versus NP is a fundamental problem in computer science and mathematics that deals with the efficiency of algorithms. It seeks to determine if every problem with a feasible solution also has an efficient solution. In simple terms, it asks whether it is easier to verify a solution (P) than to find the solution itself (NP). It is considered one of the most significant unsolved problems in computer science.

Importance: One of the most important unsolved problems in computer science

Complexity theory: P is the complexity class that represents problems solvable in polynomial time, while NP represents problems whose solutions can be verified in polynomial time

Significance: If P = NP, it would imply that many difficult problems can be solved efficiently, potentially transforming fields like cryptography, optimization, and artificial intelligence

Millennium Prize Problem: P versus NP is one of the seven unsolved problems in mathematics designated by the Clay Mathematics Institute as a Millennium Prize Problem

NP-Complete problems: If P = NP, all NP problems are also in P, including the NP-Complete problems which are considered to be the hardest problems within NP
P versus NP in other rankings
- Position 6 of 10 in Most difficult unsolved mathematical problem
7

7
votes
Birch and Swinnerton-Dyer Conjecture

This is a conjecture in number theory that relates to the behavior of elliptic curves, which are important in cryptography and other areas of mathematics. It remains unsolved and is considered one of the most important open problems in number theory.
The Birch and Swinnerton-Dyer Conjecture is one of the unsolved problems in mathematics that relates to elliptic curves, which are a type of mathematical object with fascinating properties. It was proposed by Bryan Birch and Peter Swinnerton-Dyer in 1965. The conjecture posits that there is a fundamental connection between the arithmetic properties of elliptic curves and the behavior of their associated L-functions. If the L-function of an elliptic curve has a specific value at a certain critical point, then the curve has an infinite number of rational points. On the other hand, if the L-function has a specific value of zero at that point, then the curve has only finitely many rational points. The conjecture has wide-ranging implications in number theory and algebraic geometry.

Year Proposed: 1965

Type of Problem: Unsolved Problem

Mathematical Field: Number Theory and Algebraic Geometry

Subject Area: Elliptic Curves

Connection: Arithmetic properties of elliptic curves and their L-functions
Birch and Swinnerton-Dyer Conjecture in other rankings
- Position 2 of 10 in Most difficult unsolved mathematical problem
- Position 4 of 10 in Most difficult math question
- Position 4 of 10 in Most advanced math problem
8

4
votes
AzaToth · GPL

Kepler Conjecture

Thomas C. Hales

This problem concerns the densest possible arrangement of spheres in three-dimensional space. It was proved by Thomas Hales in 1998 after a proof that relied on computer calculations.
The Kepler Conjecture is a famous problem in mathematics that deals with sphere packing in three dimensions. It attempts to determine the densest possible arrangement of identical spheres in space, without any gaps or overlaps.

Conjecture statement: No packing of congruent spheres in three dimensions has a density greater than the face-centered cubic (fcc) packing.

Number of spheres: Infinite (The conjecture aims to find the theoretical upper limit of sphere packing density).

Space dimension: Three dimensions

Type of spheres: Congruent spheres of equal size

Desired arrangement: Densest possible packing with no gaps or overlaps
Kepler Conjecture in other rankings
- Position 8 of 10 in Most difficult equation
9

3
votes
The Collatz Conjecture

This is a simple-sounding problem that asks whether a certain sequence of numbers always reaches 1, no matter what starting number is chosen. Despite its simplicity, it remains unsolved and has been the subject of much mathematical investigation.
The Collatz Conjecture is an unsolved mathematical problem that involves iteratively applying a sequence of operations to a positive integer. The conjecture states that regardless of the starting value, this sequence will always eventually reach the value 1.

Operation: If the current number is even, divide it by 2. If the current number is odd, multiply it by 3 and add 1.

Iterative Process: Repeat the operation on the resulting number.

Expected Convergence: The conjecture suggests that for any positive integer input, the sequence will eventually reach the value 1.

Open Problem: The conjecture remains unproven, despite extensive computational verification for many starting values.

Lothar Collatz Prize: In 2018, the Collatz Conjecture gained attention with the announcement of a $1 million prize for its solution.
The Collatz Conjecture in other rankings
- Position 7 of 10 in Most difficult math question
- Position 7 of 10 in Most difficult unsolved mathematical problem
10

1
votes
The Goldbach Conjecture

Christian Goldbach

This conjecture states that every even number greater than 2 can be expressed as the sum of two prime numbers. Despite being easy to state, it remains unsolved and has been the subject of much mathematical investigation.
The Goldbach Conjecture is one of the oldest unsolved problems in number theory. It states that every even integer greater than 2 can be expressed as the sum of two prime numbers.

Year of Conjecture: 1742

Conjecture Type: Unsolved Problem

Field: Number Theory

Difficulty Level: Unknown

Proof Status: Unproven
The Goldbach Conjecture in other rankings
- Position 8 of 10 in Most difficult math question

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Discussion

Ranking factors for difficult equation

Complexity

The number of variables, terms, and intricate structures in the equation determines its complexity. Highly non-linear equations and systems of equations with numerous variables are usually more challenging to solve.
Analytical vs. Numerical solving techniques

Analytical techniques aim to find an exact, general solution in a closed-form expression, whereas numerical techniques provide an approximate solution. Some equations may have no known analytical solution, which would make them inherently more difficult to solve.
Multidimensionality

Higher-dimensional equations typically involve more complex relationships between the variables and parameters, making them tougher to solve.
Nonlinearity

Equations with nonlinear terms or that require non-intuitive transformations can make them difficult to solve analytically or numerically.
Existence and uniqueness of solutions

Some equations may have no solution, a unique solution, or multiple solutions, with varying degrees of difficulty for identifying and distinguishing between them.
Coupling and interdependence

If an equation is part of a system of equations where the variables or parameters in one equation are connected to variables or parameters in another equation, it might be more challenging to solve.
Boundary and initial conditions

The difficulty of solving an equation can also depend on the specific boundary and initial conditions provided, which can change the nature of the solution.
Sensitivity to parameter changes

Some equations are highly sensitive to changes in the values of their parameters, making them difficult to solve accurately and causing small changes in inputs to produce significant differences in solutions.
Computational cost

The difficulty of an equation can also be measured in terms of the computational resources required to obtain a solution, including processing time, memory, and algorithmic complexity.
Availability of solving techniques

Some equations may be difficult to solve simply because there isn't a well-known or widely accepted method for finding a solution. The existence of specialized mathematical tools or techniques can make some equations more manageable than others.

About this ranking

This is a community-based ranking of the most difficult equation to solve. 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!

Statistics

2076 views
179 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.

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More information on most difficult equation to solve

Background Information: The Most Difficult Equation to Solve Mathematics is a subject that fascinates many people, but it can also be incredibly challenging. There are some equations that are notoriously difficult to solve, and mathematicians have been working on them for centuries. One of the most difficult equations to solve is the Riemann Hypothesis, which deals with the distribution of prime numbers. This equation was first proposed by the German mathematician Bernhard Riemann in 1859 and has never been solved. Another difficult equation is the Navier-Stokes equations, which describe the motion of fluids. These equations have been studied for almost 200 years, and while much progress has been made, they remain unsolved. Other difficult equations include the P versus NP problem, which deals with the efficiency of algorithms, and the Birch and Swinnerton-Dyer conjecture, which is related to the behavior of elliptic curves. While these equations may seem esoteric and unrelated to everyday life, they have important implications for fields such as cryptography, computer science, and physics. The quest to solve these equations has been a driving force in the development of mathematics, and mathematicians continue to work on them to this day.