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Unsolved problems in mathematics

This article lists some [unsolved problems] ** in ** [mathematics] . See individual articles for details and sources.

** 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]

- The [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 structure] s?

** 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 quadruplet] s?

- Are there infinitely many [Mersenne prime] s ( [Lenstra-Pomerance-Wagstaff conjecture] ); equivalently, infinitely many even [perfect number] s?

- Are there infinitely many [Sophie Germain prime] s?

- Are there infinitely many [regular prime] s, and if so is their relative density e^{-1/2}?

- Are there infinitely many [Cullen prime] s?

- Are there infinitely many [palindromic prime] s in base 10?

- Are there infinitely many [Fibonacci prime] s?

- Is every [Fermat number] 22 * 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]

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