Proof in Mathematics
Millenium Problems
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Search term - Millenium problem
Millennium Problem, any of seven mathematical problems designated such by the Clay Mathematics Institute (CMI) of Cambridge, Mass., U.S., each of which has a million-dollar reward for its solution. CMI was founded in 1998 by American businessman Landon T. Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.
During 2002 and 2003 Russian mathematician Grigori Perelman published three papers over the Internet that gave a “sketchy” proof of the Poincaré conjecture. His basic proof was expanded by several mathematicians and universally accepted as valid by 2006. That year Perelman was awarded a Fields Medal, which he refused. Because Perelman published his papers over the Internet rather than in a peer-reviewed journal, as required by the CMI rules, he was not offered CMI’s award, though representatives for the organization indicated that they might relax their requirements in his case. Complicating any such decision was uncertainty over whether Perelman would accept the money; he publicly stated that he would not decide until the award was offered to him. In 2010 CMI offered Perelman the reward for proving the Poincaré conjecture, and Perelman refused the money.
William L. Hosch
Encyclopædia Britannica. (n.d.). Millennium Problem. Britannica School. Retrieved August 30, 2021, from https://school.eb.com/levels/high/article/Millennium-Problem/475868
Reimann Hypothesis
Other than the “trivial zeros” along the negative real axis, all the solutions to the Riemann zeta function must lie in the critical strip of complex numbers whose real part is between 0 and 1. The Riemann hypothesis is that all these nontrivial zeros actually lie on the critical line, or Re(S) = 1/2.
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Search term - Riemann hypothesis
Riemann hypothesis, in number theory, hypothesis by German mathematician Bernhard Riemann concerning the location of solutions to the Riemann zeta function, which is connected to the prime number theorem and has important implications for the distribution of prime numbers.
The zeta function is defined as the infinite seriesζ(s) = 1 + 2−s + 3−s + 4−s + ⋯,or, in more compact notation,
where the summation (Σ) of terms for n runs from 1 to infinity through the positive integers and s is a fixed positive integer greater than 1.
A proof of the Riemann hypothesis would have far-reaching consequences for number theory and for the use of primes in cryptography.
Encyclopædia Britannica. (n.d.). Riemann hypothesis. Britannica School. Retrieved August 30, 2021, from https://school.eb.com/levels/high/article/Riemann-hypothesis/475861
P v NP
In 1971, American mathematician and computer scientist Stephen Cook (1939-) and Russian-American computer scientist Leonid Levin (1948--) independently formulated what came to be know as P versus NP problem that mathematically reduces and classifies challenges such as finding the optimal transistor array on a computer chip into a P (i.e., easy to find) versus NP (i.e., easy to check) problem.
Stephen Cook Advances Knowledge of NP-Complete Problems, Assisting Computer Scientists. (2001). In N. Schlager & J. Lauer (Eds.), Science and Its Times (Vol. 7). Gale. https://link.gale.com/apps/doc/CV2643450874/SUIC?u=61_alls&sid=bookmark-SUIC&xid=739f1ecf