Unsolved Problems in Mathematics

Erdős conjecture on arithmetic progressions

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Millennium Prize Problems

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

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

Number theory: prime numbers

General number theory

Algebraic number theory

Discrete geometry

Ramsey theory

General algebra


  • 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?

Graph theory


Partial differential equations

  • Regularity of solutions of Vlasov-Maxwell equations
  • Regularity of solutions of Euler equations

Group theory

Set theory



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