Unsolved Problems

There are many unsolved problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily so well known) include

1. The Goldbach conjecture.

2. The Riemann hypothesis.

3. The conjecture that there exists a Hadamard matrix for every positive multiple of 4.

4. The twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes).

5. Determination of whether NP-problems are actually P-problems.

6. The Collatz problem.

7. Proof that the 196-algorithm does not terminate when applied to the number 196.

8. Proof that 10 is a solitary number.

9. Finding a formula for the probability that two elements chosen at random generate the symmetric group S_n.

10. Solving the happy end problem for arbitrary n.

11. Finding an Euler brick whose space diagonal is also an integer.

12. Proving which numbers can be represented as a sum of three or four (positive or negative) cubic numbers.

13. Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers n such that phi(n)|(n-1), where phi(n) is the totient function.

14. Determining if the Euler-Mascheroni constant is irrational.

15. Deriving an analytic form for the square site percolation threshold.

16. Determining if any odd perfect numbers exist.

The Clay Mathematics Institute ( of Cambridge, Massachusetts (CMI) has named seven "Millennium Prize Problems," selected by focusing on important classic questions in mathematics that have resisted solution over the years. A $7 million prize fund has been established for the solution to these problems, with $1 million allocated to each. The problems consist of the Riemann hypothesis, Poincaré conjecture, Hodge conjecture, Swinnerton-Dyer Conjecture, solution of the Navier-Stokes equations, formulation of Yang-Mills theory, and determination of whether NP-problems are actually P-problems.

In 1900, David Hilbert proposed a list of 23 outstanding problems in mathematics (Hilbert's problems), a number of which have now been solved, but some of which remain open. In 1912, Landau proposed four simply stated problems, now known as Landau's problems, which continue to defy attack even today. One hundred years after Hilbert, Smale (2000) proposed a list of 18 outstanding problems.

K. S. Brown, D. Eppstein, S. Finch, and C. Kimberling maintain webpages of unsolved problems in mathematics. Classic texts on unsolved problems in various areas of mathematics are Croft et al. (1991), in geometry, and Guy (2004), in number theory.

See also

Beal's Conjecture, Catalan's Conjecture, Fermat's Last Theorem, Hilbert's Problems, Kepler Conjecture, Landau's Problems, Mathematics Contests, Mathematics Prizes, Poincaré Conjecture, Problem, Solved Problems, Szemerédi's Theorem, Twin Primes

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Clay Mathematics Institute. "Millennium Prize Problems.", H. T.; Falconer, K. J.; and Guy, R. K. Unsolved Problems in Geometry. New York: Springer-Verlag, p. 3, 1991.Demaine, E. D.; Mitchell, J. S. B.; and O'Rourke, J. (Eds.). "The Open Problems Project.", T. "Graphs: Theory-Algorithms-Complexity.", D. "Open Problems.", S. "Unsolved Problems.", R. K. Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, 2004.Kimberling, C. "Unsolved Problems and Rewards.", V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52, 131-145, 1979.MathPages. "Most Wanted List of Elementary Unsolved Problems.", H. Unsolved and Unsolvable Problems in Geometry. London: Oliver & Boyd, 1966.Ogilvy, C. S. Tomorrow's Math: Unsolved Problems for the Amateur, 2nd ed. New York: Oxford University Press, 1972.Ogilvy, C. S. "Some Unsolved Problems of Modern Geometry." Ch. 11 in Excursions in Geometry. New York: Dover, pp. 143-153, 1990.Ramachandra, K. "Many Famous Conjectures on Primes; Meagre But Precious Progress of a Deep Nature." Proc. Indian Nat. Sci. Acad. Part A 64, 643-650, 1998.Smale, S. "Mathematical Problems for the Next Century." Math. Intelligencer 20, No. 2, 7-15, 1998.Smale, S. "Mathematical Problems for the Next Century." In Mathematics: Frontiers and Perspectives 2000 (Ed. V. Arnold, M. Atiyah, P. Lax, and B. Mazur). Providence, RI: Amer. Math. Soc., 2000.Stephan, R. "Prove or Disprove. 100 Conjectures from the OEIS." 27 Sep 2004., R. "Do you have a comment or news on conjectures in the article math.CO/0409509?" Mill, J. and Reed, G. M. (Eds.). Open Problems in Topology. New York: Elsevier, 1990.Weisstein, E. W. "Books about Mathematics Problems.", D. "Open Problems--Graph Theory and Combinatorics.", S. "Open Problems & Projects."

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Unsolved Problems

Cite this as:

Weisstein, Eric W. "Unsolved Problems." From MathWorld--A Wolfram Web Resource.

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