# Clay institute prize Many hard problems in NP for which we don't have a jack kingston sweep floors good polynomial time algorithm are very useful.
One of the most basic facts in number theory is that any number can be broken down into a unique product of prime numbers: numbers that are not divisible by any number other than themselves and lyft promo code 2017 one.P is contained in NP: Any problem that can be solved quickly by a computer can also have a particular possible answer quickly checked by a computer.For example, every year the.Yet this has never been mathematically proven.One of the seven problems has been solved, and the other six are the subject of a great deal of current research.For historical and technical reasons, problems where we can quickly check a possible solution are said to be solvable in "nondeterministic polynomial time or "NP.".Clay, mathematics, institute, millennium, prize, problems, seven problems judged to be among the most important open questions in mathematics.In the world of math and computer science, there are a lot of problems that we know how to program a computer to solve "quickly" basic arithmetic, sorting a list, searching through a data table.Here's a simple example with four cities and their distances marked in blue (distances how do you play minute to win it are clearly not to scale Business Insider/Andy Kiersz, here's one possible tour our salesman could take through the four cities: Business Insider/Andy Kiersz, start at A, walk to B (which has.P stands for "polynomial time An algorithm like residency matching can be run in a number of steps based on a power of the size of the input.This makes answering the million-dollar question somewhat easier: You either need to find an efficient algorithm for one NP complete problem, or prove that no such algorithm exists for one particular such problem.This means that if there is an efficient polynomial time algorithm for NP complete problems, large numbers can be factored quickly, and internet security based on RSA or similar protocols would fall apart.On the other hand, finding a proof that no such algorithms exist, and that P NP, would likely involve a huge leap in our understanding of the nature and limitations of computers.Easy Problems, there are many useful problems that are in P: We know how to solve them relatively quickly, even for large inputs.There are algorithms for the Traveling Salesman Problem that are much more efficient than this brute-force approach, but they all either provide some kind of approximate "good enough" solution that might not be the actual shortest path, or still have the number of needed calculations.Being able to definitively make such a statement about these kinds of problems would likely require a much deeper understanding of the nature of information and computation than we currently have, and would almost certainly have profound and far-reaching consequences.For small numbers, finding factorizations is easy: 15 is 5 3; 12 is 2 2 3; 70 is 2.NP, and answering the question would earn you a million-dollar prize.Finding efficient algorithms for the hard problems in NP, and showing that P NP, would dramatically change the world.To see why, consider the most naive possible way to go about factoring a number: Take every number between 2 and one less than the number, and divide the number we're interested in by the smaller number.

The basic idea is fairly straightforward.
As we can already see with these small numbers of cities, the number of paths grows extremely quickly as we add more cities.

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