The Most Difficult Equation, Ranked

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

Updated on Apr 25, 2024 06:24

Navigating the complexities of mathematical equations can often feel like solving a puzzle without the edges. For students and professionals alike, understanding which equations pose the greatest challenge can be a stepping stone to deeper learning. Ranking these formidable equations not only sheds light on where collective difficulties lie but also catalyzes discussions on how best to approach and solve them. This dynamic ranking system invites users from all backgrounds to contribute their experiences and insights by voting for the equations they find most intricate. It acts as a communal database where knowledge and struggles are shared, making the opaque walls of advanced mathematics more penetrable for everyone. With each vote, the list is recalibrated, reflecting current perceptions and understanding in real-time.

What Is the Most Difficult Equation?

1

83
votes
Navier-Stokes Equations

These equations describe the motion of fluids, but they are notoriously difficult to solve due to their complexity and nonlinear nature. They are so challenging that solving them is one of the seven unsolved Millennium Prize Problems.
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluid substances such as liquids and gases.

Equations: The equations consist of the continuity equation and the momentum equations.

Continuity equation: Describes the conservation of mass for an incompressible fluid.

Momentum equations: Describe the conservation of momentum for a fluid, taking into account pressure and viscous forces.

Nonlinear nature: The equations are nonlinear, making them difficult to solve analytically.

Important in fluid dynamics: They provide a mathematical description of the behavior of fluids in various scenarios.
Navier-Stokes Equations in other rankings
- Position 1 of 10 in Most difficult math equation
- Position 3 of 10 in Most difficult math question
- Position 4 of 10 in Most difficult unsolved mathematical problem
- Position 5 of 10 in Most advanced math problem
- Position 8 of 10 in Most advanced mathematical proof used in reasoning
2

61
votes
Riemann Hypothesis

Bernhard Riemann

This equation is related to prime numbers and is considered one of the most important unsolved problems in mathematics. It has eluded mathematicians for over a century.
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 3 of 10 in Most advanced mathematical proof used in reasoning
- Position 4 of 10 in Most difficult equation to solve
3

25
votes
P vs. NP Problem

This equation relates to the complexity of algorithms, and it remains unsolved despite being a central problem in computer science.
The P vs. NP problem is one of the most famous unsolved problems in computer science and mathematics. It deals with the question of whether every problem for which a solution can be verified quickly can also be solved quickly. In simplified terms, it asks if it is easier to verify a solution than to find one. This problem has widespread implications in fields such as cryptography, optimization, and artificial intelligence. If P (problems solvable in polynomial time) is indeed equal to NP (problems verifiable in polynomial time), it would mean that many difficult problems could be solved efficiently, revolutionizing various industries. However, proving or disproving this conjecture remains a major challenge and has eluded researchers for several decades.

Problem Type: The problem belongs to the field of computational complexity theory.

P: P refers to the class of problems that can be solved in polynomial time.

NP: NP refers to the class of problems for which a solution can be verified in polynomial time.

Goal: The goal is to determine whether P is equal to NP or not.

Millennium Prize Problem: The Clay Mathematics Institute has named the P vs. NP problem as one of the Millennium Prize Problems, offering a prize of $1 million for its resolution.
4

19
votes
Fermat's Last Theorem

Pierre de Fermat

This theorem was famously unsolved for over 350 years until Andrew Wiles finally proved it in 1994. It relates to the equation xn + yn = zn and whether there are any whole number solutions for n greater than 2.
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 3 of 10 in Most difficult equation to solve
- Position 4 of 10 in Most difficult math equation
- Position 5 of 10 in Most difficult math question
5

14
votes
Hilbert's Tenth Problem

David Hilbert

This problem relates to Diophantine equations, which are equations that seek integer solutions. It was shown to be unsolvable by Yuri Matiyasevich in 1970.
Hilbert's Tenth Problem, also known as the Entscheidungsproblem (decision problem), was formulated by the German mathematician David Hilbert in 1900. It asks whether there is an algorithmic method to determine whether a given polynomial equation with integer coefficients has integer solutions.

Difficulty: One of the most difficult problems in mathematics

Unsolvability: Proven to be unsolvable by Yuri Matiyasevich in 1970

Connection to Turing's Work: Inspired Alan Turing to formulate the concept of a universal Turing machine

Diophantine Equations: Specifically deals with Diophantine equations, which involve polynomial equations with integer coefficients

Undecidability: Demonstrates the concept of undecidable problems, which cannot be solved algorithmically
6

16
votes
Yang-Mills Theory

This equation describes the behavior of subatomic particles, and it is notoriously difficult to solve due to its complexity.
The Yang-Mills Theory is a mathematical framework that describes the behavior of elementary particles in the context of Quantum Field Theory. It provides a way to understand the strong nuclear force, electromagnetism, and weak nuclear force as different aspects of a single mathematical structure.

Gauge invariance: The theory is invariant under local gauge transformations.

Field equations: The theory is described by a set of nonlinear differential equations.

Quantum chromodynamics: Describes interactions between quarks and gluons.

Yang-Mills potential: The gauge field is represented by a set of potentials.

Lie algebra: The symmetry group of the theory is described by a Lie algebra.
Yang-Mills Theory in other rankings
- Position 9 of 10 in Most advanced mathematical proof used in reasoning
7

2
votes
The Birch and Swinnerton-Dyer Conjecture

This equation relates to the properties of elliptic curves, and it remains unsolved despite being a central problem in number theory.
The Birch and Swinnerton-Dyer Conjecture is a major unsolved problem in number theory, specifically in algebraic number theory and modular forms. It deals with the relationship between the solutions of certain elliptic curves and the behavior of their associated L-functions. The conjecture was first stated in 1965.

Status: Unsolved

Field: Number theory

Focus: Algebraic number theory, modular forms

Statement: Relates the rank of an elliptic curve to the order of its zero at s=1

Importance: One of the seven Clay Mathematics Institute Millennium Prize Problems
The Birch and Swinnerton-Dyer Conjecture in other rankings
- Position 5 of 10 in Most difficult math equation
8

5
votes
AzaToth · GPL

Kepler Conjecture

Thomas C. Hales

This conjecture relates to the densest possible arrangement of spheres in three dimensions, and it was proven by Thomas C. Hales in 1998.
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 to solve
9

7
votes
Lorenz Equations

Edward Lorenz

These equations describe the behavior of chaotic systems, and they are difficult to solve due to their sensitivity to initial conditions.
The Lorenz Equations is a system of nonlinear ordinary differential equations that exhibit chaotic behavior. It was derived by the meteorologist Edward Lorenz in 1963 while studying convection in the Earth’s atmosphere. The Lorenz Equations have since become one of the most well-known examples of chaos theory and have applications in various fields such as physics, mathematics, and engineering.

System of Equations: The Lorenz Equations consist of three coupled differential equations.

Nonlinear Dynamics: The equations are nonlinear and exhibit sensitive dependence on initial conditions.

Lorenz Attractor: The solutions to the Lorenz Equations often produce the Lorenz Attractor, a set of chaotic trajectories in phase space.

Butterfly Effect: The Lorenz Equations are famous for demonstrating the butterfly effect, where small changes in initial conditions lead to vastly different outcomes.

Parameter Values: The equations are typically defined with specific parameter values: σ (sigma), ρ (rho), and β (beta).
10

10
votes
Sbyrnes321 · CC0

Schrödinger Equation

Erwin Schrödinger

This equation describes the behavior of quantum systems, and it is notoriously difficult to solve due to its complexity and the need for advanced mathematical techniques such as linear algebra and partial differential equations.
The Schrödinger Equation is a fundamental equation in quantum mechanics that describes the behavior of quantum systems. It determines the wave function of a system and how it evolves over time.

1: First proposed by Erwin Schrödinger in 1926

2: Written as a partial differential equation

3: Describes the energy levels and wave functions of particles in quantum systems

4: Applies to a wide range of systems, from atoms and molecules to subatomic particles

5: Non-relativistic equation, does not take into account relativistic effects

Missing your favorite equation?

Graphs

Discussion

Ranking factors for difficult equation

Complexity

The complexity of an equation is often determined by the number of variables, terms, and mathematical operations involved. Equations with higher complexity tend to be more difficult to understand and solve.
Abstractness

The level of abstractness may impact the difficulty of an equation, as more abstract concepts require a deeper understanding of the underlying mathematical principles.
Non-linearity

Non-linear equations often require more advanced mathematical techniques and may involve more complex relations between variables, making them more difficult to solve compared to linear equations.
Existence and uniqueness of solutions

The difficulty of an equation can be affected by whether a solution exists and, if so, whether the solution is unique. Equations without a well-defined solution or with multiple solutions can be more challenging to work with.
Numerical stability

The difficulty of an equation can also be influenced by its numerical stability, which refers to the sensitivity of the solution to small changes in the input data.
Pre-requisites

The level and number of pre-requisite mathematical concepts and techniques needed to understand and solve the equation should also be considered. Equations requiring advanced mathematical knowledge are typically more difficult.
Real-world applicability

Equations that are more closely related to real-world problems tend to be more difficult, as they may require the incorporation of additional factors, assumptions, or constraints.
Analytic vs. numerical approaches

The availability of analytic solutions (i.e., solutions that can be explicitly derived using algebraic or calculus techniques) versus numerical solutions (i.e., solutions that require computational methods) can influence the difficulty of an equation.
Multidisciplinary connections

Equations that involve concepts from multiple disciplines, such as physics, engineering, and computer science, may be more difficult, as they require a broader range of knowledge and skills.
Intuition

The degree to which an equation is counterintuitive or difficult to visualize can also impact its difficulty.

About this ranking

This is a community-based ranking of the most difficult 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!

Statistics

3272 views
236 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.

More information on most difficult equation

Background Information: What is the Most Difficult Equation? Mathematics is a fascinating subject that has intrigued human minds for centuries. It is a tool that has allowed us to understand the world around us, from the smallest particles to the largest galaxies. However, some equations have proven to be more challenging than others, leaving even the brightest minds scratching their heads. So, what is the most difficult equation? The answer to this question is not so straightforward. There are many complex equations out there, each with its own set of challenges. Some equations require extensive knowledge of calculus, while others are based on abstract mathematical concepts that are difficult to grasp. One equation that is often cited as being particularly difficult is the Navier-Stokes equation. This equation describes the motion of fluids and is used in a wide range of fields, from engineering to meteorology. However, despite its importance, mathematicians have yet to find a general solution to this equation, making it one of the most challenging problems in mathematics. Another equation that is often mentioned is the Riemann hypothesis. This equation is related to the distribution of prime numbers and has important implications for cryptography and computer security. However, despite decades of research, mathematicians have yet to prove or disprove the hypothesis. Overall, the most difficult equation is a matter of debate and depends on the context in which it is being applied. However, one thing is for sure – these complex equations continue to challenge and inspire mathematicians around the world.