So is P equal to NP? Scientists disagree

From a recent edition of the great “Why Is This Interesting” newsletter:

P vs. NP is one of the great unsolved problems in cs/math and is one of the seven Clay Institute Millenium Problems with a million-dollar bounty. To get that million, all you have to do is prove that there’s a difference (or not) between problems that are easy for computers to solve (P) and those that are hard (NP). If that sounds a little crazy, it’s because it is. Nearly every person working on P vs. NP believes that P is not the same as NP—that there is some fundamental distinction between what is easy and hard for a computer to solve—but to this point, no one has proven this is true.

Most things we ask a computer to do are relatively straightforward: sorting a list of words alphabetically, for example, can happen nearly instantaneously even if that list is 20,000 words long. These are called “P” problems because a regular computer can solve them in polynomial time. Polynomial-time is most easily understood as the number of elements you’re processing—say words in a list that need to be sorted—raised to a fixed power like n^2 or n^5.

On the other hand, some problems are more challenging for computers to solve: cryptographic techniques like finding prime factors, playing games like sudoku, solving math/logic questions like the traveling salesman problem, and even finding the right configuration of packing boxes in your trunk, are all “hard” in CS terms. In other words, we don’t have an efficient algorithm to solve them.

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