Question about density of prime numbers?

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SUMMARY

The discussion centers on the density of prime numbers, specifically addressing Bertrand's Postulate, which asserts that there is always at least one prime number between any integer n and 2n. As n increases, the average distance between consecutive prime numbers grows, yet the number of primes in the interval from n to 2n remains substantial. The density of primes near n can be approximated by the formula 1/log(n), leading to an estimate of the number of primes between n and 2n as n/log(n).

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  • Understanding of prime numbers and their distribution
  • Familiarity with Bertrand's Postulate
  • Basic knowledge of logarithmic functions
  • Concept of density in number theory
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  • Study the implications of Bertrand's Postulate in number theory
  • Explore the distribution of prime numbers using the Prime Number Theorem
  • Learn about the Riemann Hypothesis and its relation to prime density
  • Investigate advanced techniques for estimating prime gaps
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Mathematicians, number theorists, and students interested in the properties and distribution of prime numbers will benefit from this discussion.

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It is known that prime numbers become sparser and sparser, with the average distance between one prime number and the next increasing as n approaches infinity. Dividing an even number by 2 results in a bottom half from 1 to n / 2 and a top half from n / 2 to n. For a particular sufficiently large n, would the top half from n / 2 to n no longer contains any prime numbers?
 
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No, there is always a prime between n and 2n (which is the same thing as between n/2 and n). This is called Bertrand's Postulate, but despite the name, it's a theorem.

In fact, there are a LOT of primes in that interval, and the bigger n is, the more there are. The density of the primes near n is like ##{1 \over \log n}##, so a rough estimate for the number of primes between n and 2n is ##{n \over \log 2n} \approx {n \over \log n}##.
 

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