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Everything about Unsolved Problems In Mathematics totally explained

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 ones yet to be solved are:

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
  • 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 e^
  • infinitely many Cullen primes
  • infinitely many palindromic primes in base 10
  • infinitely many Fibonacci primes
  • Is every Fermat number 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

    General number theory

  • abc conjecture
  • existence of odd perfect numbers
  • existence of quasiperfect numbers
  • existence of odd weird numbers
  • existence of Lychrel numbers
  • Proof that 10 is a solitary number
  • Solving the Happy Ending problem for arbitrary n
  • existence of Taxicab(5, 2, n) for n>1.

    Ramsey theory

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

    General algebra

  • Hilbert's sixteenth problem
  • Hadamard conjecture
  • existence of perfect cuboids
  • Existence of quadratic number fields being Euclidean but not norm-Euclidean

    Combinatorics

  • Number of Magic squares
  • Finding a formula for the probability that two elements chosen at random generate the symmetric group S_n

    Graph theory

  • Erdős-Gyárfás conjecture
  • The Hadwiger conjecture
  • The Ringel-Kotzig conjecture
  • 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)

    Analysis

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

    Group theory

  • Is every finitely presented periodic group finite?
  • The inverse Galois problem

    Other

  • Generalized star height problem
  • Invariant subspace problem
  • Modeling black hole mergers
  • Problems in Latin squares
  • Problems in loop theory and quasigroup theory

    Problems solved recently

  • Road coloring conjecture (Avraham Trahtman, 2007)
  • The Angel problem (Various independent proofs, 2006)
  • Stanley-Wilf conjecture (Gabor Tardos and Adam Marcus, 2004)
  • Poincaré conjecture (Solution by Grigori Perelman in 2002 now confirmed)
  • Catalan's conjecture (Preda Mihăilescu, 2002)
  • Kato's conjecture (Auscher, Hofmann, Lacey, and Tchamitchian, 2001)
  • The Langlands program for function fields (Laurent Lafforgue, 1999)
  • Taniyama-Shimura conjecture (Wiles, Breuil, Conrad, Diamond, and Taylor, 1999)
  • Kepler conjecture (Thomas Hales, 1998)
  • Milnor conjecture (Vladimir Voevodsky, 1996)
  • Fermat's last theorem (Andrew Wiles, 1994)
  • Bieberbach conjecture (Louis de Branges, 1985)
  • Four color theorem (Appel and Haken, 1977)Further Information

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