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Counting Primes (3/19)

Counting Primes (3/19). Given a number n , how many primes are there between 2 and n? No one has discovered an exact formula (and no one will!). So, change the question: Given a number n , about how many primes are there between 2 and n?

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Counting Primes (3/19)

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  1. Counting Primes (3/19) • Given a number n, how many primes are there between 2 and n? • No one has discovered an exact formula (and no one will!). • So, change the question: Given a number n, about how many primes are there between 2 and n? • Let’s experiment a bit with Mathematica. We denote the exact number of primes below n by (n). • The Prime Number Theorem (PNT). The number of primes below n is approximated by n / ln(n). More specifically:

  2. Comments on the PNT • It was a huge accomplishment of 19th Century mathematics. • Another (illuminating) way to say what the PNT says is that in the neighborhood of a number n, about 1 out of every ln(n) numbers will be primes. Or, put another way, the density of primes near n is 1 / ln(n). • This leads us to an even better estimator for (n): the “logarithmic integral” Li(n) = • Check this out in Mathematica .

  3. More Comments • It’s absolutely astounding (to me at least) that the number of primes below n is somehow related to the number e 2.71828. • What is this number e anyway? Where does it come from? Where does it arise in nature? • Well, it’s most easily described as the natural limit of compounding, i.e., . • For Friday, please read Chapter 13.

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