The prime number theorem is a statement of a quantuplicity: there are about N/log N prime numbers in the first N numbers.
quantuplicity noun. The quantitative relationship between two amounts. Usually referring specifically to the case when this is expressed as a ratio giving the number of times that one contains another (or vice versa.)
Calculating quantuplicities for different sociological quantities can lead one to a very complicated poset.
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In my last math post I casually mentioned that the sum of the reciprocals of the primes diverges. That is
\[
\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\frac{1}{13}+ \dots=\infty
\]
That seems like a hard thing to prove. Certainly none of the traditional convergence tests…
We've spent more than a few Sunday Function features discussing the properties of the prime numbers. They're just so important and interesting in number theory that they're an irresistible target. Let's set some scenery before getting to the actual function this week.
There are an infinite number…
We're doing two functions today. If I'm not mistaken we've done each of them separately, but there's a famous and interesting relationship between the two that's always interesting to look at. Like very many interesting mathematical facts, it has to do with the prime numbers.
As such the first…
In the same basic vein as last week's How to Read a Scientific Paper, here's a kind of online draft of the class I'm going to give Friday on the appropriate ways to present scientific data. "Present" here meaning the more general "display in some form, be it a talk, a poster, a paper, or just a…
You know this whole prime number thing has piqued my interest since those two Mersenne primes were found last week (or whenever). I'm teaching Real Analysis this semester for the math department and we've been talking about infinite sequences. Did you know, while no one has proven the Mersenne's are infinite, a couple of guys have conjectured that they are not only infinite but they follow some sort of an exponential form. This quantuplicity thing gives me the idea of potentially comparing Mersenne primes to regular primes (which Euclid actually proved were infinite).
Hmmm... And it's Saturday night and what am I doing? Replying to a blog post about prime numbers. Such is life when one is married with children. Ok, off to read some more Murakami...
Dave, speaking of words, the Alibi crossword this week had as a clue: "California town with an accidentally palindromic bakery."