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> This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random

by c3534l 11y ago

> This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. I don't think this is true at all. Take a look at the famous Ulam Spiral: http://scienceblogs.com/goodmath/wp-content/blogs.dir/476/files/2012/04/i-8e4cdfc0a83e388851408bd4b44fd1e4-Sacks%20spiral.png http://scienceblogs.com/goodmath/wp-content/blogs.dir/476/fi... You can see that while prime numbers are difficult to predict, they're anything but random. I'm not sure why the article is claiming that mathematicians used to think the distribution of primes was evenly distributed, which is complete and utter nonsense.

mjn 11y ago
At a high level it's not that far off, in the sense that most mathematicians think nontrivial patterns in the primes are at least unusual, and they mostly behave randomly. But it's true that there is some structure, which is in jargon terms called a "conspiracy" among the primes when it's found or hypothesized. As Terence Tao summarizes it, > We believe that the primes do not observe any significant pattern beyond the obvious ones (e.g. mostly being odd), but we are still a long way from making this belief completely rigorous. That's from this set of slides on structure and randomness in the primes, which has some other relevant bits in it: https://terrytao.files.wordpress.com/2009/07/primes1.pdf https://terrytao.files.wordpress.com/2009/07/primes1.pdf Especially relevant are slides 10-11 on treating the primes as a pseudorandom set, and then slides 14-15 on using pseudorandom models of the primes to rigorously (vs. heuristically) prove theorems. That's done by classifying and ruling out all possible ways nonrandom structure in the actual primes (the "conspiracies") could sink the specific theorem being proven.
Intermernet 11y ago
Ulam spirals are maddening. I've played with them for years now, and the patterns are so close to predictable, but always break upon close inspection. I have a feeling that prime theory, and Ulam spirals in particular, will drive many mathematicians slightly crazy if they dwell on them too long. On the other hand, they create wonderful patterns. I'm getting a laser-cut Ulam spiral done to hang on my wall :-)

by c3534l 11y ago

mjn 11y ago

At a high level it's not that far off, in the sense that most mathematicians think nontrivial patterns in the primes are at least unusual, and they mostly behave randomly. But it's true that there is some structure, which is in jargon terms called a "conspiracy" among the primes when it's found or hypothesized. As Terence Tao summarizes it, > We believe that the primes do not observe any significant pattern beyond the obvious ones (e.g. mostly being odd), but we are still a long way from making this belief completely rigorous. That's from this set of slides on structure and randomness in the primes, which has some other relevant bits in it: https://terrytao.files.wordpress.com/2009/07/primes1.pdf https://terrytao.files.wordpress.com/2009/07/primes1.pdf Especially relevant are slides 10-11 on treating the primes as a pseudorandom set, and then slides 14-15 on using pseudorandom models of the primes to rigorously (vs. heuristically) prove theorems. That's done by classifying and ruling out all possible ways nonrandom structure in the actual primes (the "conspiracies") could sink the specific theorem being proven.

Intermernet 11y ago

Ulam spirals are maddening. I've played with them for years now, and the patterns are so close to predictable, but always break upon close inspection. I have a feeling that prime theory, and Ulam spirals in particular, will drive many mathematicians slightly crazy if they dwell on them too long. On the other hand, they create wonderful patterns. I'm getting a laser-cut Ulam spiral done to hang on my wall :-)