Millennium Prize Problems - Win $1M Prize Solving This Maths Problems

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Millennium Prize Problems - Win $1M Prize Solving This Maths Problems Millennium Prize Problems - Win $1,000,000 Prize by Solving This Mathematics Problems It may strike many as odd that there are unfinished bits of mathematics.

Millennium Prize Problems - Win $1,000,000 Prize Money by Solving This Mathematics Problem. In school, maths is presented as a completed textbook that was worked out hundreds of years ago. The truth is that mathematics is a constantly developing field: some problems can be solved comparatively easily while others seem completely resistant to attack. One such problem is known as Goldbach's conjecture and it has confounded mathematicians since 1742. It involves prime numbers, which are numbers that do not have any factors other than 1 and themselves. (Some of my favourites include 7, 31 and 8,191.) Goldbach said in a letter to his friend Euler that it looks like all even numbers are the sum of two prime numbers. In the two and a half centuries since, no one has found a single even number that cannot be written as the sum of two prime numbers. Go on: try a few. I did.

16 = 5 + 11

40 = 17 + 23

31,556,926 = 89,459 + 31,467,467

I've only tried a handful but other people have gone further. Even before calculators and computers, mathematicians had manually checked every even number up to 100,000. As soon as this arithmetic grinding could be done by electrons, progress exploded; as it stands all of the even numbers up to a billion billion (1.6 x 1018 to be exact) have been checked. Which is officially a lot. If you had started checking those numbers at the rate of one a second from the beginning of the universe 13.7bn years ago, you would only now be about half way. Surely we can call off the hunt? Goldbach's conjecture is clearly true, isn't it? The trouble is that mathematicians need to know for sure that there will never be an even number that doesn't work. For every number you check, there is always a bigger one that could fail the test. A variation of Fermat's Last Theorem that looked certain to be true, failed on the number 61,917,364,224. Just checking numbers is never enough: showing that something will always, undoubtedly work is known as a proof in maths and a problem isn't finished until one is found. Which is not to say that unproven mathematical concepts are not useful. Our modern technological world is based on all sorts of mathematical concepts and methods, some proven and some unproven but very well tested. Unproven concepts are used every day by mathematicians, engineers, medical researchers, programmers and indeed all of us via the technology at our fingertips. But their unstable, proof-less foundations leave mathematicians feeling a bit queasy.

https://www.youtube.com/watch?v=yhtcJPI6AtY
02/11/2016

https://www.youtube.com/watch?v=yhtcJPI6AtY

Mathematics is patterns and logic, imagination and rigor. It is a way of seeing and a way of thinking. Math Mornings is a series of public lectures aimed at ...

Millennium Prize ProblemsRiemann hypothesishttp://www.claymath.org/millenium-problems/riemann-hypothesisRules for the Mi...
01/10/2014

Millennium Prize Problems
Riemann hypothesis
http://www.claymath.org/millenium-problems/riemann-hypothesis

Rules for the Millennium Prizes
http://www.claymath.org/millennium-problems/rules-millennium-prizes

Related Documents:
Official Problem Description by Enrico Bombieri
http://www.claymath.org/sites/default/files/official_problem_description.pdf

The Riemann Hypothesis by Peter Sarnak
http://www.claymath.org/sites/default/files/sarnak_rh_0.pdf

Lecture by Jeff Vaaler
http://claymath.msri.org/riemannhypothesis.mov

Millennium Prize ProblemsYang–Mills existence and mass gaphttp://www.claymath.org/millenium-problems/yang%E2%80%93mills-...
01/10/2014

Millennium Prize Problems
Yang–Mills existence and mass gap
http://www.claymath.org/millenium-problems/yang%E2%80%93mills-and-mass-gap

Rules for the Millennium Prizes
http://www.claymath.org/millennium-problems/rules-millennium-prizes

Related Documents:
Official Problem Description by Arthur Jaffe and Edward Witten
http://www.claymath.org/sites/default/files/yangmills.pdf

Status of the Problem by Michael Douglas
http://www.claymath.org/sites/default/files/ym2.pdf

Lecture by Lorenzo Sadun
http://claymath.msri.org/yangmillsexistence.mov

01/10/2014

Millennium Prize Problems http://en.wikipedia.org/wiki/Millennium_Prize_Problems
Offer by Clay Mathematics Institute http://en.wikipedia.org/wiki/Clay_Mathematics_Institute

P versus NP problem http://en.wikipedia.org/wiki/P_versus_NP_problem

Hodge conjecture http://en.wikipedia.org/wiki/Hodge_conjecture

Poincaré conjecture (solved) http://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture

Riemann hypothesis http://en.wikipedia.org/wiki/Riemann_hypothesis

Yang–Mills existence and mass gap http://en.wikipedia.org/wiki/Yang%E2%80%93Mills_existence_and_mass_gap

Navier–Stokes existence and smoothness http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_existence_and_smoothness

Birch and Swinnerton-Dyer conjecture http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture

Doubts continue on claim to have solved P vs NP mathematical questionAug 17, 2010One of the most complex mathematical pr...
01/10/2014

Doubts continue on claim to have solved P vs NP mathematical question
Aug 17, 2010

One of the most complex mathematical problems in the world is proving either that P ≠ NP or P=NP, a riddle that was first formulated in 1971 by mathematicians Leonid Levin and Stephen Cook. The question was one of seven millennium problems set by the Clay Mathematical Institute (CMI) in Cambridge, Massachusetts as being among the most difficult to solve.

Vinay Deolalikar, a computer scientist from Hewlett-Packard's research arm based in Palo Alto, California, claims to have solved the problem and stands to win $1 million if his proof is accepted.
P refers to problems with solutions that are easy to find, in other words P is a class of problems for which an answer can be found in polynomial time. NP refers to problems with solutions that are almost impossible to find, but easy to verify, or in other words a class of problems for which an answer can be verified in polynomial time.
The Clay Mathematical Institute offers the following problem to illustrate NP: you have a list of 400 university students to be accommodated but there is only room in the dormitory for 100 students. The Dean has provided a list of pairs of incompatible students who cannot be on the final list. This is an NP-problem because it is extremely difficult to go through all possible combinations to come up with a list, but very easy to verify once a list is obtained. According to CMI the number of possible combinations is greater than the number of atoms in the universe.
If P ≠ NP some complex problems, such as the one above, could never be solved efficiently and there are serious limits on what computers will be able to accomplish in the future, whereas if P = NP every such problem would have a polynomial time solution. Mr Deolalikar writes that this would have profound implications for applications such as cryptography and on the question of whether or not human creativity could be automated.
Among those skeptical of Deolalikar's claim is Scott Aaronson, assistant professor of electrical engineering and computer science at the Massachusetts Institute of Technology (MIT), who said on his blog that he will personally add $200,000 to the prize, even though he can ill afford it, and he has not read Deolalikar's entire paper. He said if the proof was correct it would change his life so dramatically that "having to pay $200,000 will be the least of it." He later also gave his reasons for being so confident in MIT's technology review.
Another skeptic is Professor Richard Lipton at Georgia Tech College of Computing, who has been working on the P vs NP problem for three decades.
Deolalikar's paper is available online, and is being peer-reviewed by computer scientists. To be considered for the million dollar prize the paper must be published in an approved mathematics publication and must also retain general acceptance by mathematicians two years after the proof is published.

More information:
http://phys.org/news201232319.html

One of the most complex mathematical problems in the world is proving either that P ≠ NP or P=NP, a riddle that was first formulated in 1971 by mathematicians Leonid Levin and Stephen Cook. The question was one of seven millennium problems set by the Clay Mathematical Institute (CMI) in Cambridge, ....

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