In science and mathematics, an open problem or an open question is a known problem which can be accurately stated, and which is assumed to have an objective and verifiable solution, but which has not yet been solved (no solution for it is known).
In the history of science, some of these supposed open problems were "solved" by means of showing that they were not, after all, well-defined. Many open problems in mathematics are in fact concerned with the question whether a certain definition is or is not consistent.
Two notable examples in mathematics that have been solved and closed by researchers in the late twentieth century are Fermat's Last Theorem and the four color theorem. An important open mathematics problem solved in the early 21st century is the Poincaré conjecture.
Important open problems exist in all scientific fields. For example, one of the most important open problems in biochemistry is the protein structure prediction problem – how to predict a protein's structure from its sequence.
- Faltings, Gerd (July 1995), "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF), Notices of the AMS, 42 (7): 743–746, ISSN 0002-9920
- K. Appel and W. Haken (1977), "Every planar map is four colorable. Part I. Discharging", Illinois J. Math 21: 429–490. MR 58:27598d
- K. Appel, W. Haken, and J. Koch (1977), "Every planar map is four colorable. Part II. Reducibility", Illinois J. Math 21: 491–567. MR 58:27598d
- Vendruscolo, M.; Najmanovich, R.; Domany, E. (1999), "Protein Folding in Contact Map Space", Physical Review Letters, 82 (3): 656–659, arXiv: , Bibcode:1999PhRvL..82..656V, doi:10.1103/PhysRevLett.82.656
- Dill, K.A.; Ozkan, S.B.; Weikl, T.R.; Chodera, J.D.; Voelz, V.A. (2007), "The protein folding problem: when will it be solved?" (PDF), Current Opinion in Structural Biology, 17 (3): 342–346, doi:10.1016/j.sbi.2007.06.001, PMID 17572080
- Kennedy, Donald; Norman, Colin (2005), "What Don't We Know?", Science, 309 (5731): 75–75, doi:10.1126/science.309.5731.75, PMID 15994521
- "So much more to know", Science, 309 (5731): 78–102, July 2005, doi:10.1126/science.309.5731.78b, PMID 15994524
- Open Problem Garden The collection of open problems in mathematics build on the principle of user editable ("wiki") site