Category Archives: open problems

Book review: Fermat’s Enigma

Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical ProblemSimon Singh After having it recommended to me several times, I finally picked up this book when I happened to see it at our favorite local used bookstore. I … Continue reading

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Book Review: The Enigma of the Spiral Waves

The Enigma of the Spiral Waves (Secrets of Creation Volume 2)words by Matthew Watkins, pictures by Matt Tweed Matthew Watkins and Matt Tweed have done it again! I previously wrote a (very positive) review of Volume I—this book is just … Continue reading

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Book review: In Pursuit of the Traveling Salesman

As mathematical problems go, the “traveling salesman problem” (TSP) is a rare gem: it is simultaneously of great theoretical, historical, and practical interest. On the theoretical front, it is a well-known example of the class of “NP-complete” problems, which lie … Continue reading

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17×17 4-coloring with no monochromatic rectangles

Quick, what’s special about the following picture? As just announced by Bill Gasarch, this is a grid which has been four-colored (that is, each point in the grid has been assigned one of four colors) in such a way that … Continue reading

Posted in open problems, pattern, people, pictures | Tagged , , , , , | 5 Comments

Collatz conjecture: an apology

In the comments on my previous post about Gerhard Opfer’s proposed proof of the Collatz conjecture, several different people expressed the opinion that my tone was rather arrogant, and I think they have a point. So: I apologize for my … Continue reading

Posted in meta, open problems, proof | 9 Comments

The Collatz conjecture is safe (for now)

A few days ago John Cook reported a draft paper claiming to solve the Collatz conjecture. Of course, since the Collatz conjecture is so simple to state, it constantly attracts tons of would-be solvers, and most of the purported “proofs” … Continue reading

Posted in open problems, proof | 43 Comments

P vs NP: What’s the problem?

As promised (better late than never), I’m going to begin explaining the (in)famous P vs NP question (see the previous post for a bit more context). As a start, here’s a super-concise, 30,000-foot version of the question: Are there problems … Continue reading

Posted in computation, open problems | Tagged , , , , | 9 Comments

P ≠ NP?

A few days ago, Vinay Deolalikar, a Principal Research Scientist at HP Labs, began circulating a draft of a paper entitled “P ≠ NP”. The mathematics and computer science communities immediately erupted in a frenzy of excitement and activity. The … Continue reading

Posted in computation, links, open problems, people, proof | Tagged , , , | 7 Comments

A gentle introduction to the 5th Polymath project

I highly recommend reading Jason Dyer’s description of the Erdős discrepancy problem, the subject of the most recent Polymath project (the Polymath projects are an experiment in massively collaborative mathematics, where anyone at all can contribute something towards a solution). … Continue reading

Posted in arithmetic, counting, links, number theory, open problems | Tagged , ,

Perfect numbers, part III

This is the last in a series of posts about perfect numbers. A quick recap of the series so far: in part I, I defined perfect numbers as positive integers n for which , where denotes the sum of the … Continue reading

Posted in algebra, number theory, open problems, primes, solutions | 11 Comments