Millennium Prize Problems
- P versus NP
- The Hodge conjecture
- Riemann Hypothesis
- Yang–Mills existence and mass gap
- Navier-Stokes existence and smoothness
- The Birch and Swinnerton-Dyer conjecture.
Only the Poincaré conjecture has been solved. The smooth four dimensional Poincaré conjecture is still unsolved. That is, can a four dimensional topological sphere have two or more inequivalent smooth structures?
Other still-unsolved problems
Additive number theory
- Goldbach’s conjecture and its weak version
- The values of g(k) and G(k) in Waring’s problem
- Collatz conjecture (3n + 1 conjecture)
- Gilbreath’s conjecture
Number theory: prime numbers
- Catalan’s Mersenne conjecture
- Twin prime conjecture
- Are there infinitely many prime quadruplets?
- Are there infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers?
- Are there infinitely many Sophie Germain primes?
- Are there infinitely many regular primes, and if so is their relative density e − 1 / 2?
- Are there infinitely many Cullen primes?
- Are there infinitely many palindromic primes in base 10?
- Are there infinitely many Fibonacci primes?
- Are there any Wall-Sun-Sun primes?
- Is every Fermat number 22n + 1 composite for n > 4?
- Is 78,557 the lowest Sierpinski number?
- Is 509,203 the lowest Riesel number?
- Fortune’s conjecture (that no Fortunate number is composite)
- Polignac’s conjecture
- Landau’s problems
- Does every prime number appear in the Euclid-Mullin sequence?
General number theory
- abc conjecture
- Do any odd perfect numbers exist?
- Do quasiperfect numbers exist?
- Do any odd weird numbers exist?
- Do any Lychrel numbers exist?
- Is 10 a solitary number?
- Do any Taxicab(5, 2, n) exist for n>1?
- Brocard’s problem: existence of integers, n,m, such that n!+1=m2 other than n=4,5,7
- Distribution and upper bound of mimic numbers
Algebraic number theory
- Solving the Happy Ending problem for arbitrary n
- Finding matching upper and lower bounds for K-sets and halving lines
- The Hadwiger conjecture on covering n-dimensional convex bodies with at most 2n smaller copies
- Number of Magic squares (sequence A006052 in OEIS)
- Finding a formula for the probability that two elements chosen at random generate the symmetric group Sn
- Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets
- The Lonely runner conjecture: if k + 1 runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be more than a distance 1 / (k + 1)from each other runner) at some time?
- Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?
- The Erdős-Gyárfás conjecture on cycles with power-of-two lengths in cubic graphs
- The Hadwiger conjecture relating coloring to clique minors
- The Erdős–Faber–Lovász conjecture on coloring unions of cliques
- The total coloring conjecture
- The list coloring conjecture
- The Ringel-Kotzig conjecture on graceful labeling of trees
- The Hadwiger–Nelson problem on the chromatic number of unit distance graphs
- Deriving a closed-form expression for the percolation threshold values, especially pc (square site)
- Tutte’s conjectures that every bridgeless graph has a nowhere-zero 5-flow and every bridgeless graph without the Petersen graph as a minor has a nowhere-zero 4-flow
- The Reconstruction conjecture and New digraph reconstruction conjecture concerning whether or not a graph is recognizable by the vertex deleted subgraphs.
- The cycle double cover conjecture that every bridgeless graph has a family of cycles that includes each edge twice.
- Does a Moore graph with girth 5 and degree 57 exist?
- the Jacobian conjecture
- Schanuel’s conjecture
- Lehmer’s conjecture
- Pompeiu problem
- Is γ (the Euler-Mascheroni constant) irrational?
- the Khabibullin’s conjecture on integral inequalities
Partial differential equations
- Regularity of solutions of Vlasov-Maxwell equations
- Regularity of solutions of Euler equations
- Is every finitely presented periodic group finite?
- The inverse Galois problem
- For which positive integers m, n is the free Burnside group B(m,n) finite? In particular, is B(2, 5) finite?
- Is there a simple group which is not hypertranssimple?
- The problem of finding the ultimate core model, one that contains all large cardinals.
- If ℵω is a strong limit cardinal, then 2ℵω < ℵω1 (see Singular cardinals hypothesis). The best bound, ℵω4, was obtained by Shelah using his pcf theory.
- Woodin’s Ω-hypothesis.
- Does the consistency of the existence of a strongly compact cardinal imply the consistent existence of a supercompact cardinal?
- (Woodin) Does the Generalized Continuum Hypothesis below a strongly compact cardinal imply the Generalized Continuum Hypothesis everywhere?
- Does there exist a Jonsson algebra on ℵω?
- Without assuming the axiom of choice, can a nontrivial elementary embedding V→V exist?
- Is it consistent that ?
- Does the Generalized Continuum Hypothesis entail for every singular cardinal λ?
- Generalized star height problem
- Invariant subspace problem
- Problems in Latin squares
- Problems in loop theory and quasigroup theory
- Dixmier conjecture
- Excerpted: Pereleman, Poincaré and two types of Math (hypios.com)
- A New Million Dollar Prize? (rjlipton.wordpress.com)
- Russian mathematician Grigory Perelmanturned down $1 million for proving the Poincaré conjecture. What’s that? (slate.com)
- ICM2010 – Smirnov laudatio (gowers.wordpress.com)
- A Generalization of Erdos’s Proof of Bertrand-Chebyshev Theorem (kintali.wordpress.com)
- P is not equal to NP, does it really matter? (globalthoughtz.com)
- The Poincaré Conjecture Explained (laplacian.wordpress.com)
- Mysteries of Math: Unsolved Problems & Unexplained Patterns (weburbanist.com)