Unsolved Problems in Math & Science: from Riemann and Collatz to P vs NP

Thank you for visiting this site. This article presents a guided list of the famous unsolved problems of mathematics and science.

“Unsolved problems” may conjure discussions meaningful only to specialists. In fact, several of mathematics’ famous open problems have statements a schoolchild can understand. Can every even number be written as two primes? Does the game of tripling and adding one always come back to 1? The rules are as simple as play — yet the world’s geniuses have attacked them for centuries, and not one person has produced a proof. That gap, I think, is this genre’s greatest attraction.

Nor are these mere brain-teasers. Some carry a one-million-dollar bounty per problem, and the list includes questions wired straight into modern society’s foundations — the security of internet encryption, and the limits of what computers can do.

This article organizes the famous unsolved problems into 4 genres and 15 problems: “primes and integers,” “simple rules and computation,” “shapes and equations,” and “physics and the cosmos.” All seven “Millennium Prize Problems” — mathematics’ most important questions, at a million dollars each — are included. Every problem has its own full article.

The Conjectures That Challenge Primes and Integers

The first genre: the unsolved problems of primes and integers, the royalty of mathematics. That the primes 2, 3, 5, 7, 11… continue forever has been known for 2,300 years — yet when it comes to their “arrangement,” even modern mathematics has not reached the core. Each problem’s statement fits in a line.

The Riemann hypothesis has stood unsolved since 1859 — over 160 years; Goldbach, since 1742 — over 280. Computer verification finds Goldbach counterexample-free out to the staggering figure of four quintillion. And still, in the world of mathematics, no pile of examples ever proves “true for all numbers.”

The Mysteries of Simple Rules and Computation

The next genre concerns “operations” and “computation.” The Collatz conjecture is a bottomless mystery born of a plaything’s rule. P vs NP is a question with computer science’s very existence staked on it — and one of the million-dollar Millennium problems.

Collatz is the problem that made the genius Erdős say “mathematics is not yet ready for such problems.” If P vs NP resolves in the affirmative, much of internet encryption collapses on paper. The gap between innocent appearance and monumental consequence is best savored in the full articles.

The Mysteries of Physics and the Cosmos

The third genre leaps out of mathematics into the real universe itself. Flowing water, orbiting bodies, the contents of the cosmos — all utterly commonplace, all with gaping holes at the center of our understanding. The Navier–Stokes problem is a Millennium Prize Problem, with a million dollars of its own.

Physical mysteries have a flavor mathematical conjectures lack: the “answer key” can arrive suddenly, by experiment or observation. The chance that tomorrow’s experiment unmasks dark matter or dark energy is not zero — and if it happens, the textbooks are rewritten overnight. Yang–Mills is a Millennium problem; dark energy is dark matter’s counterpart and the largest mystery in the universe.

The Mysteries of Shapes and Equations

The fourth genre: problems abstract yet central to mathematics, hiding deep within shapes and equations. Honestly, this is the series’ most technical territory — even catching the atmosphere takes effort. Still, the Poincaré conjecture has the tactile pull of telling the universe’s shape, and BSD offers a surprise entrance through a thousand-year-old right-triangle riddle. All three are Millennium Prize Problems (Poincaré being the lone solved one).

All Seven Millennium Prize Problems, Collected

This series has a specialty the other knowledge series lack: prize money.

In 2000, the Clay Mathematics Institute in the United States selected mathematics’ seven most important problems as the “Millennium Prize Problems,” staking one million dollars on each. This series provides a full article for every one of the seven: the Riemann hypothesis, P vs NP, Navier–Stokes, Poincaré, BSD, Yang–Mills, and Hodge, all introduced above. Of the seven, exactly one is solved — the Poincaré conjecture, proven by the Russian mathematician Perelman. Who then declined the money.

Why does a mathematics institute — not a corporation, not a government — stack up that kind of cash? Because these problems are pressure points where the map of mathematics redraws itself the instant they fall. An unsolved problem’s value lies less in the answer than in the new tools born along the way. Fermat’s Last Theorem took 360 years, falling in 1995 — and the campaign’s by-products became the principal machinery of modern number theory. “The problems that can’t be solved are what move mathematics forward” — that paradox is the fun of this genre.

Summary

This article introduced mathematics’ famous unsolved problems in one guided list.

A one-line problem statement, and no one can solve it. That gap teaches us that humanity still stands at the barest threshold of the world of numbers. Each full article pushes to the frontier of “what is known, and what is not” — start with whichever problem hooked you.

This site also publishes companion collections in the same format: the intuition-defying “logic and probability puzzles” and the “paradoxes of the world.” After tasting the vastness of unsolvable problems, returning to the catharsis of solvable ones is highly recommended.

There are also the “cognitive bias guide” on the quirks of thinking, and the “thought experiments” collection of laboratory-of-the-mind classics — companions for an intellectual pub crawl.

Unsolved mysteries live outside mathematics too. For undeciphered books and vanished people, examined with a fact-checking eye, see the “world mysteries” collection.

World Mysteries: 20 Famous Enigmas Explained, from the Voynich Manuscript to Bermudaen.senkohome.com/mystery-list/

Thank you for reading. We hope to see you in the next article.