On 8 August 1900, David Hilbert presented a list of unsolved mathematical problems to the International Congress of Mathematicians in Paris, proposing 10 in his actual lecture and 23 in the subsequently published paper. The list shaped the direction of 20th-century mathematics more than perhaps any other single document, inspiring generations of mathematicians. Some problems were resolved within years; a few remain open even today.
In 1900, German mathematician David Hilbert — widely regarded as one of the most influential mathematicians of his era — stood before the International Congress of Mathematicians in Paris and posed a question: what are the most important unsolved problems facing mathematics as it enters a new century? His answer, a list of 23 carefully chosen problems, turned out to shape the actual direction of mathematical research for the next hundred years.
The problems spanned an enormous range of mathematics: foundations of geometry, number theory, algebra, mathematical physics, and the foundations of mathematics itself. Some asked for proofs of specific, well-defined conjectures; others were more open-ended, calling for entirely new theories or asking foundational questions that didn't yet have a fully precise mathematical formulation.
Of the 23 problems, roughly half have been fully and definitively resolved. Several others have been partially resolved or resolved for important special cases. A handful remain genuinely and completely open even today, over 125 years later — including the 8th problem, which encompasses the still-unsolved Riemann Hypothesis.
| # | Problem (brief) | Status |
|---|---|---|
| 1 | Continuum Hypothesis | Shown independent of ZFC (Gödel 1940, Cohen 1963) |
| 2 | Consistency of arithmetic axioms | Gödel's 2nd Incompleteness Theorem (1931) shows this can't be settled as originally envisioned |
| 3 | Equal volume, equal-decomposability of polyhedra | Solved (Dehn, 1900) — the same year, in the negative |
| 7 | Transcendence of certain numbers (e.g., 2^√2) | Solved (Gelfond-Schneider theorem, 1934) |
| 8 | Riemann Hypothesis (and related prime distribution problems) | Unsolved |
| 10 | Algorithm for solving Diophantine equations | Proven impossible (Matiyasevich, 1970, building on Davis, Putnam, Robinson) |
| 18 | Kepler Conjecture (sphere packing) as a sub-question | Solved (Hales, 1998, computer-assisted) |
Asks whether there is a set whose cardinality (size) is strictly between that of the integers and that of the real numbers. Gödel (1940) showed it cannot be disproven from the standard ZFC axioms; Paul Cohen (1963) showed it also cannot be proven from them — meaning it is genuinely independent of the standard axioms of set theory, a landmark result in mathematical logic.
Asked for a general algorithm to determine whether any given Diophantine equation (a polynomial equation seeking integer solutions) has a solution. The eventual answer, established through the combined work of Martin Davis, Hilary Putnam, Julia Robinson, and finally Yuri Matiyasevich (1970), was that no such general algorithm can exist — a striking result connecting number theory directly to computability theory and the theoretical limits of algorithms.
Part of this problem concerned the most efficient way to pack identical spheres in space. The Kepler Conjecture (that the familiar "stacking cannonballs" arrangement is optimal) was proven by Thomas Hales in 1998, using an extensive computer-assisted proof — and was only fully, formally verified via machine-checked formal proof methods in 2014, illustrating again how some of the most difficult modern proofs increasingly require computational verification alongside traditional mathematical argument.
Hilbert's 1900 address reflected his deeply held belief, later crystallised in what became known as "Hilbert's Programme," that mathematics should in principle be able to resolve any well-posed mathematical question — a conviction he memorably expressed with the phrase "Wir müssen wissen, wir werden wissen" ("We must know, we will know"), later inscribed on his tombstone. This optimistic vision was significantly complicated (though not entirely refuted in every respect) by Gödel's Incompleteness Theorems just three decades later, which showed that some mathematical questions are provably undecidable within any single sufficiently powerful, consistent formal system.
Notably, Hilbert's original 1900 list did not include what would later become some of the most famous problems in 20th and 21st century mathematics, including the P vs NP problem (first formulated in the 1970s) or Fermat's Last Theorem (though this had already been open for over 250 years by 1900, Hilbert chose not to include it on his own list). This illustrates that even history's most prescient and mathematically sophisticated problem-selectors cannot fully anticipate which specific questions will define subsequent generations of mathematical research and interest.
In 2000, exactly a century after Hilbert's address, the Clay Mathematics Institute announced seven "Millennium Prize Problems," each carrying a $1 million reward, explicitly modelled on and inspired by Hilbert's influential precedent. As of 2026, only one of these seven (the Poincaré Conjecture, solved by Perelman in 2003) has been resolved — the remaining six, including the Riemann Hypothesis and P vs NP, remain unsolved.
Stephen Smale (1998) and others have since proposed their own updated lists of important open problems for the 21st century, explicitly following Hilbert's influential format and precedent, though none has achieved quite the same broad, sustained, and historically enduring cultural and mathematical influence as Hilbert's original 1900 list.