Imagine trying to solve the hardest problem of mathematics in the world. There are some problems that have baffled the best of the mathematicians in the world. ScienceStruck Staff Last Updated:
However, the best known quantum algorithm for this problem, Shor's algorithmdoes run in polynomial time, although this does not indicate where the problem lies with respect to non-quantum complexity classes.
Does P mean "easy"? Quadratic fit suggests that empirical algorithmic complexity for instances with 50—10, variables is O log n 2.
It is a common and reasonably accurate assumption in complexity theory; however, it has some caveats. First, it is not always true in practice.
A theoretical polynomial algorithm may have extremely large constant factors or exponents thus rendering it impractical. There are algorithms for many NP-complete problems, such as the knapsack problemthe traveling salesman problem and the Boolean satisfiability problemthat can solve to optimality many real-world instances in reasonable time.
The empirical average-case complexity time vs. An example is the simplex algorithm in linear programmingwhich works surprisingly well in practice; despite having exponential worst-case time complexity it runs on par with the best known polynomial-time algorithms.
A key reason for this belief is that after decades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than important known NP-complete problems see List of NP-complete problems.
These algorithms were sought long before the concept of NP-completeness was even defined Karp's 21 NP-complete problemsamong the first found, were all well-known existing problems at the time they were shown to be NP-complete. It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easy to verify matches real-world experience.
There would be no special value in "creative leaps," no fundamental gap between solving a problem and recognizing the solution once it's found.
For example, in these statements were made: This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration.
VardiRice University Being attached to a speculation is not a good guide to research planning. One should always try both directions of every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whose solution was opposite to their expectations, even though they had developed all the methods required.
Either direction of resolution would advance theory enormously, and perhaps have huge practical consequences as well.
It is also possible that a proof would not lead directly to efficient methods, perhaps if the proof is non-constructiveor the size of the bounding polynomial is too big to be efficient in practice. The consequences, both positive and negative, arise since various NP-complete problems are fundamental in many fields.
Cryptography, for example, relies on certain problems being difficult. A constructive and efficient solution [Note 2] to an NP-complete problem such as 3-SAT would break most existing cryptosystems including: Existing implementations of public-key cryptography a foundation for many modern security applications such as secure financial transactions over the Internet.
Cryptographic hashing as the problem of finding a pre-image that hashes to a given value must be difficult in order to be useful, and ideally should require exponential time. On the other hand, there are enormous positive consequences that would follow from rendering tractable many currently mathematically intractable problems.A math problem posed to year-olds in Singapore is creating a wave of confusion for those trying to solve it online.
A math problem for year-olds is stumping the world. Share. Tweet. Share. Online shopping from a great selection at Apps & Games Store. Ian Stewart is a math genius. For us non-geniuses, he tells us about the hardest math problem in the world.
Get the latest Chicago Bears news, scores, stats, standings, rumors, and more from the Chicago Sun-Times. The Hardest Math Problem in the World - See the Believe Does your head start spinning at the mere sight of equations and calculators? Imagine trying to solve the hardest problem of mathematics in .
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