There are many unsolved
problems in mathematics. Some prominent outstanding unsolved problems (as well as some which are not necessarily
so well known) include
3. The conjecture that there exists a
for every positive multiple of 4.
twin prime conjecture (i.e., the conjecture that there are an infinite number of twin primes).
5. Determination of whether
NP-problems are actually
7. Proof that the
196-algorithm does not terminate
when applied to the number 196.
8. Proof that 10 is a
9. Finding a formula for the probability that two elements chosen at random generate the
symmetric group .
10. Solving the
happy end problem for arbitrary .
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
Lehmer's Mahler measure problem and Lehmer's totient problem on the existence
of composite numbers such that , where is the totient function.
14. Determining if the
constant is irrational.
15. Deriving an analytic form for the square site
16. Determining if any odd
perfect numbers exist.
The Clay Mathematics Institute (
http://www.claymath.org/millennium/) 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
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
Fermat's Last Theorem
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References Clay Mathematics Institute. "Millennium Prize Problems." http://www.claymath.org/millennium/. Croft,
H. T.; Falconer, K. J.; and Guy, R. K. New York: Springer-Verlag, p. 3, 1991. Unsolved
Problems in Geometry. Demaine,
E. D.; Mitchell, J. S. B.; and O'Rourke, J. (Eds.). "The Open
Problems Project." http://cs.smith.edu/~orourke/TOPP/. Emden-Weinert,
T. "Graphs: Theory-Algorithms-Complexity." http://people.freenet.de/Emden-Weinert/graphs.html. Eppstein,
D. "Open Problems." http://www.ics.uci.edu/~eppstein/junkyard/open.html. Finch,
S. "Unsolved Problems." http://www.mathsoft.com/mathsoft_resources/unsolved_problems/. Guy,
R. K. New York: Springer-Verlag, 2004. Unsolved
Problems in Number Theory, 3rd ed. Kimberling,
C. "Unsolved Problems and Rewards." http://faculty.evansville.edu/ck6/integer/unsolved.html. Klee,
V. "Some Unsolved Problems in Plane Geometry." Math. Mag. 52,
131-145, 1979. MathPages. "Most Wanted List of Elementary Unsolved
Problems." http://www.mathpages.com/home/mwlist.htm. Meschkowski,
H. London: Oliver & Boyd, 1966. Unsolved
and Unsolvable Problems in Geometry. Ogilvy,
C. S. New York: Oxford University
Press, 1972. Tomorrow's
Math: Unsolved Problems for the Amateur, 2nd ed. Ogilvy, C. S. "Some Unsolved Problems of Modern
Geometry." Ch. 11 in New York: Dover, pp. 143-153, 1990. Excursions
in Geometry. 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 (Ed. V. Arnold, M. Atiyah, P. Lax,
and B. Mazur). Providence, RI: Amer. Math. Soc., 2000. Mathematics:
Frontiers and Perspectives 2000 Stephan,
R. "Prove or Disprove. 100 Conjectures from the OEIS." 27 Sep 2004. http://www.arxiv.org/abs/math.CO/0409509/. Stephan,
R. "Do you have a comment or news on conjectures in the article math.CO/0409509?"
Mill, J. and Reed, G. M. (Eds.). New York: Elsevier, 1990. Open
Problems in Topology. Weisstein, E. W.
"Books about Mathematics Problems." http://www.ericweisstein.com/encyclopedias/books/MathematicsProblems.html. West,
D. "Open Problems--Graph Theory and Combinatorics." http://www.math.uiuc.edu/~west/openp/. Wolfram,
S. "Open Problems & Projects." http://www.wolframscience.com/openproblems/NKSOpenProblems.pdf. Referenced
on Wolfram|Alpha Unsolved Problems
Cite this as:
Weisstein, Eric W. "Unsolved Problems."
From --A Wolfram Web Resource. MathWorld https://mathworld.wolfram.com/UnsolvedProblems.html