🔢 Mathematics Mysteries

Is every problem whose solution can be quickly verified also quickly solvable? This fundamental question about computational complexity could revolutionize computer science and cryptography.

Do smooth solutions to the Navier-Stokes equations exist globally in time, or can they develop singularities? This problem is fundamental to understanding fluid dynamics and turbulence.

Do all non-trivial zeros of the Riemann zeta function have real part equal to 1/2? This hypothesis is central to understanding the distribution of prime numbers.

Can every Hodge class on a non-singular complex algebraic variety be expressed as a linear combination of classes of algebraic cycles? This bridges algebraic geometry and topology.

Is the rank of the group of rational points on an elliptic curve equal to the order of vanishing of its L-function at s=1? This connects number theory and algebraic geometry.

Can every even integer greater than 2 be expressed as the sum of two primes? This simple-to-state problem has resisted proof for nearly 300 years despite extensive computational verification.

Are there infinitely many pairs of primes that differ by 2? Examples include (3,5), (5,7), (11,13), (17,19). This fundamental question about prime distribution remains open.

Starting with any positive integer, repeatedly apply: if even, divide by 2; if odd, multiply by 3 and add 1. Does this sequence always reach 1? Simple to understand, impossible to prove.

Can any map drawn on a plane be colored with at most four colors such that no two adjacent regions share the same color? Proved by computer in 1976, but no human-verifiable proof exists.

Is there a set whose cardinality is strictly between that of the integers and the real numbers? Gödel and Cohen proved this is independent of ZFC set theory - it can neither be proved nor disproved.