Language versions of problems are not complete. Please send relevant PDF files to the webmaster: [email protected] .

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

13. Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers such that , where is the totient function .

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 actually P-problems .

Mathematics (from Greek μάθημα * máthēma* , "knowledge, study, learning") is the study of such topics as quantity , [1] structure , [2] space , [1] and change . [3] [4] [5] There are many views among mathematicians and philosophers as to the exact scope and definition of mathematics . [6] [7]

Rigorous arguments first appeared in Greek mathematics , most notably in Euclid 's * Elements * . Since the pioneering work of Giuseppe Peano (1858–1932), David Hilbert (1862–1943), and others on axiomatic systems in the late 19th century , it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions . Mathematics developed at a relatively slow pace until the Renaissance , when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day. [10]

Mathematics is essential in many fields, including natural science , engineering, medicine, finance and the social sciences . Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory . Mathematicians also engage in pure mathematics , or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. [16]

Language versions of problems are not complete. Please send relevant PDF files to the webmaster: [email protected] .

Language versions of problems are not complete. Please send relevant PDF files to the webmaster: [email protected] .

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

13. Lehmer's Mahler measure problem and Lehmer's totient problem on the existence of composite numbers such that , where is the totient function .

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 actually P-problems .

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