Questões em aberto em Matemática

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Millennium Prize Problems
Of the seven Millennium Prize Problems set by the Clay Mathematics Institute, the six ones yet to be solved are:
P versus NP
The Hodge Conjecture
The Riemann Hypothesis
Yang-Mills Existence and Mass Gap
Navier-Stokes existence and smoothness
The Birch and Swinnerton-Dyer Conjecture

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
Twin prime conjecture
infinitely many prime quadruplets
infinitely many Mersenne primes (Lenstra-Pomerance-Wagstaff conjecture); equivalently, infinitely many even perfect numbers
infinitely many regular primes, is their density
infinitely many Cullen primes
infinitely many palindromic primes in base 10
Is every Fermat number composite for n > 4?
Is 78,557 the lowest Sierpinski number?
Is 509,203 the lowest Riesel number?

General number theory
abc conjecture
existence of odd perfect numbers
existence of quasiperfect numbers
existence of odd weird numbers
Proof that the 196-algorithm does not terminate when applied to the number 196
Proof that 10 is a solitary number
Solving the Happy Ending problem for arbitrary n

Ramsey theory
The values of the Ramsey numbers, particularly R(5,5)
The values of the Van der Warden numbers

General algebra
Hilbert's sixteenth problem
Hadamard conjecture
existence of perfect cuboids

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 Sn

Graph theory
Erdős-Gyárfás conjecture
The Graph isomorphism problem
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)

Analysis
Schanuel's conjecture
Lehmer's conjecture
Pompeiu problem
Is γ (the Euler-Mascheroni constant) irrational?

Group theory
Is every finitely presented periodic group finite?

Other
Generalized star height problem
Invariant subspace problem
Modeling black hole mergers
The Angel problem