Mathematics is a field filled with mysteries that have challenged the brightest minds for centuries. Some problems have been cracked after relentless effort, while others remain unsolved, teasing mathematicians with their complexity. These legendary puzzles range from simple-sounding statements that hide deep truths to questions that could revolutionize entire fields. Let’s dive into some of the most famous solved and unsolved mathematical problems that have shaped history.
Fermat’s Last Theorem: The 350-Year Puzzle
Pierre de Fermat scribbled a deceptively simple note in 1637, claiming he had a proof that no three positive integers \( a \), \( b \), and \( c \) satisfy \( a^n + b^n = c^n \) for \( n > 2 \). He famously added that his proof wouldn’t fit in the margin, leaving mathematicians baffled for centuries. British mathematician Andrew Wiles finally cracked it in 1994 after decades of work. His proof stretched over 100 pages and used advanced concepts like elliptic curves. This theorem is legendary not just for its difficulty but also for how it pushed mathematical boundaries.
The Riemann Hypothesis: Prime Number Enigma
Bernhard Riemann’s 1859 hypothesis suggests that all non-trivial zeros of the zeta function lie on a specific line in the complex plane. This seemingly abstract idea holds the key to understanding prime numbers, which are the building blocks of mathematics. Solving it could revolutionize cryptography and number theory. The Clay Mathematics Institute has even offered a $1 million reward for a proof. Despite intense efforts, no one has cracked it yet, making it one of math’s most famous open questions.
The Collatz Conjecture: A Child’s Game That Stumps Geniuses
Take any positive integer: if it’s even, divide by 2; if it’s odd, multiply by 3 and add 1. Repeat the process, and the Collatz Conjecture claims you’ll always reach 1. It sounds like a simple arithmetic game, yet no one has proven it works for every number. Mathematician Paul Erdős famously said math “isn’t ready” for this problem. Despite extensive testing—it holds true for numbers up to trillions—a general proof remains elusive.
The Four Color Theorem: A Computer’s Triumph
Can any map be colored with just four colors so no two adjacent regions match? This question, posed in 1852, was finally answered in 1976—but controversially. The proof relied heavily on computer calculations, a first in major mathematics. Purists argued that a traditional handwritten proof was needed, sparking debates about what counts as valid math. Nevertheless, the theorem stands, proving that sometimes machines can solve what humans alone cannot.
The Twin Prime Conjecture: Endless Pairs?
Are there infinitely many prime pairs differing by 2, like (11, 13) or (17, 19)? This is the heart of the Twin Prime Conjecture. In 2013, Yitang Zhang shocked the math world by proving that primes infinitely often appear within 70 million of each other. While this narrowed the gap, the original conjecture—whether the gap can be as small as 2—remains unproven. Mathematicians continue to chip away at this tantalizing problem, inching closer to an answer.
The P vs NP Problem: A Million-Dollar Question
This problem asks whether every question with a quickly verifiable solution also has a quick solving method. Think of it like puzzles: if verifying an answer is easy, is finding one just as easy? Solving P vs NP could transform computing, cryptography, and logistics. It’s another Millennium Prize problem with a $1 million reward, but despite decades of work, no one knows the answer.
Goldbach’s Conjecture: Even Numbers as Prime Sums
Proposed in 1742, this conjecture states that every even integer greater than 2 can be written as the sum of two primes. For example, 4 = 2 + 2, 10 = 3 + 7. Computers have verified it for numbers up to unfathomable sizes, but a general proof remains out of reach. It’s one of those problems that seems obvious yet defies all attempts at a solution.
The Navier-Stokes Equations: Fluid Dynamics Mystery
These equations describe how fluids like air and water move, yet mathematicians still don’t fully understand them. Proving whether smooth solutions always exist is another Millennium Prize challenge. Weather prediction, aerodynamics, and even medicine rely on these equations, making their resolution crucial for science and engineering.
The Poincaré Conjecture: Solved After a Century
Henri Poincaré’s 1904 question about the shape of the universe was finally answered in 2003 by Grigori Perelman. He proved that in three dimensions, a simply connected compact space must be a sphere. Perelman famously declined the $1 million prize, adding to the intrigue around this once-mysterious problem.
The Hodge Conjecture: Geometry’s Deep Puzzle
This conjecture links algebraic geometry and topology, asking whether certain shapes can be built from simpler ones. It’s a central question in modern math but remains unsolved, resisting even the sharpest geometric minds. Its resolution would unify major areas of mathematics, making it a holy grail for theorists.
The Birch and Swinnerton-Dyer Conjecture
This problem connects the number of solutions to elliptic curves with a special function’s behavior at a critical point. It’s another Millennium Prize problem with deep implications for number theory. Despite progress, the full conjecture remains unproven, teasing mathematicians with its elegance and complexity.
The Kissing Number Problem: Spheres Touching Spheres
How many identical spheres can touch a central sphere without overlapping? In 2D, it’s 6 (like coins around a central coin). In 3D, it’s 12, but higher dimensions get murky. While solved for some cases, the general problem remains open, blending geometry and combinatorics in a uniquely tactile challenge.
The Hadwiger Conjecture: Graph Theory’s Big Question
This conjecture predicts a deep relationship between graph coloring and graph minors. It’s been proven for small cases but remains unverified in general, standing as one of graph theory’s most important unsolved problems. If true, it would reveal fundamental truths about networks and structures.
The Erdős Discrepancy Problem: Randomness or Pattern?
Paul Erdős wondered whether infinite sequences must contain hidden patterns. The problem was solved in 2015, showing that even seemingly random sequences have unavoidable structures. This victory, though recent, shows how even old puzzles can still yield to modern techniques.
Mathematics thrives on these challenges, blending history, mystery, and innovation. Each problem, solved or unsolved, represents a frontier of human curiosity and intellect.
Besides founding Festivaltopia, Luca is the co founder of trib, an art and fashion collectiv you find on several regional events and online. Also he is part of the management board at HORiZONTE, a group travel provider in Germany.
