MathsClub keeps the ledger of deep mathematics: the great unsolved problems, the ones AI has just felled — and the centuries-old conjectures that finally fell to human minds. In 2026 alone, machines disproved Erdős's unit-distance conjecture and toppled the 87-year-old Jacobian conjecture. The frontier is moving.
PROBLEM OF THE WEEK
Navier–Stokes Existence and Smoothness openDo solutions of the equations that describe fluid motion always exist and stay smooth, or can they blow up? Mathematics has not caught up with the water in the glass.
Read the full entry →The frontier moved here — newest resolutions first.
Berge guessed perfection means no odd holes; forty years later Chudnovsky–Robertson–Seymour–Thomas proved it in ~150 pages. The capstone of perfection.
Jaeger's summit conjecture: every bridgeless cubic graph maps into the Petersen graph — implying cycle double cover, Berge–Fulkerson, and five-flow at once.
Colouring vertices and edges together: is maximum degree plus two always enough? Behzad asked in 1965; the exact bound is still open.
Does fractional edge-colouring pin down integral edge-colouring up to one colour? Goldberg and Seymour guessed yes in the 1970s; Chen–Jing–Zang proved it in 2019.
Are the nth roots of the primes strictly decreasing? Firoozbakht's 1982 guess would control prime gaps far beyond Legendre — verified computationally, proved nowhere.
How large can the intersection of two free subgroups be? Hanna Neumann asked in 1957; Mineyev and Friedman proved the sharp bound independently in 2011.
