World Solve

This weaker cousin of Goldbach's Conjecture illustrates how even modest-sounding claims about primes can remain unproven for centuries. It highlights how little we still understand about the raw arithmetic of prime numbers.

Mersenne primes, of the form two raised to a power minus one, are the largest primes ever discovered, but no one has proven that the list never ends. Finding new ones remains an active computational hunt.

If pi is what mathematicians call 'normal,' every string of digits, including your birthday or an entire book encoded in numbers, would appear infinitely often. No proof exists either way, despite pi's digits being computed to trillions of places.

This difficulty is exactly what keeps most modern encryption secure, since multiplying primes together is easy but reversing the process is not. If an efficient method were found, it could upend the security of the internet as we know it.

Sphere-packing problems are solved for a few specific dimensions, but the general pattern across all possible dimensions remains unknown. This affects fields ranging from error-correcting codes to theoretical physics.

Despite chess being fully deterministic with no hidden information, its complexity is so vast that no one has proven who truly holds the advantage under perfect play. Solving smaller games has offered clues, but full chess remains unresolved.

Mathematicians have many tools for distinguishing knots, but no single method works perfectly and efficiently for every possible case. This unsolved problem sits at the heart of the mathematical field of knot theory.

Starting from any positive number and repeatedly applying a basic rule always seems to eventually reach the number one, but no proof confirms this holds for every possible starting point. Its simplicity contrasted with its difficulty has made it famous among mathematicians and hobbyists alike.

Certain simple-sounding tiling and packing puzzles remain surprisingly resistant to a complete general solution. These problems test the limits of combinatorics and geometric reasoning.

This question, closely related to the once-open Poincaré Conjecture, has been resolved in three dimensions but opens further mysteries in higher-dimensional analogues. Exploring these generalizations remains an active area of topology.

Some conjectures suggest that most true statements in sufficiently complex systems may have no short proof at all, only extremely long or even infinite ones. This unresolved question challenges our basic assumptions about what makes mathematics knowable.

The famous four-color theorem was resolved for flat maps, but generalizing the problem to more complex or higher-dimensional surfaces remains unsettled in several cases. This continues to be explored in graph theory and topology.

Some sequences and structures appear chaotic without ever settling into repetition, and proving whether repetition is truly avoidable in specific systems remains unresolved. This connects deeply to dynamical systems and number theory.

Simple geometric fairness questions like this remain surprisingly open for many classes of shapes, especially in three or more dimensions. Solving them has practical relevance to resource division and computational geometry.

Primes appear scattered with no obvious repeating pattern, and finding a true predictive formula, rather than just an approximation, remains unsolved. This is closely tied to the deeper mysteries of the Riemann Hypothesis.

Classifying all finite simple groups was a monumental achievement, but fully understanding how all groups relate to and build from these simple pieces remains an ongoing task. This shapes our foundational understanding of symmetry itself.

Quantum computers show promise for specific problems like factoring, but whether they offer a genuine speedup across all hard computational problems remains unproven. This question sits at the intersection of mathematics, physics, and computer science.

True randomness is difficult to define and even harder to prove, since any formula-based sequence is, by definition, predictable if you know the formula. This unresolved tension underlies debates in both mathematics and philosophy of probability.

While the standard cube's maximum was determined through massive computation, generalized versions with more dimensions or larger sizes remain computationally and theoretically unresolved. This connects group theory with combinatorial optimization.

This boolean satisfiability problem is central to computer science, and no efficient general method is known for solving every possible case quickly. It remains a benchmark problem for understanding computational complexity.