## Millennium Prize Problems

Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six yet to be solved are:

- 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 (3
*n*+ 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 2
^{2}^{n}+ 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=*m*^{2}other than*n*=4,5,7 - Distribution and upper bound of mimic numbers

### Algebraic number theory

- Are there an infinite number of real quadratic number fields with unique factorization?

### Discrete geometry

- 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 2^{n}smaller copies

### Ramsey theory

- The values of the Ramsey numbers, particularly
*R*(5,5) - The values of the Van der Waerden numbers

### General algebra

### Combinatorics

- Number of Magic squares (sequence A006052 in OEIS)
- Finding a formula for the probability that two elements chosen at random generate the symmetric group
*S*_{n} - 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?

### Graph theory

- 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
*p*_{c}(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?

### Analysis

- 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

### Group theory

- 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?

### Set theory

- The problem of finding the ultimate core model, one that contains all large cardinals.
- If ℵ
_{ω}is a strong limit cardinal, then 2^{ℵω}< ℵ_{ω}_{1}(see Singular cardinals hypothesis). The best bound, ℵ_{ω}_{4}, 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 ℵ
_{ω}? - 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?

### Other

- Generalized star height problem
- Invariant subspace problem
- Problems in Latin squares
- Problems in loop theory and quasigroup theory
- Dixmier conjecture

###### (Source: Wikipedia)

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