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Greg Martin
Greg Martin
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I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This leads us to believe through probabilistic models that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the laymanlayperson.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.

I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This leads us to believe through probabilistic models that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the layman.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.

I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This leads us to believe through probabilistic models that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the layperson.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.

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Tim Ratigan
Tim Ratigan
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I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This is essentially derived by assumingleads us to believe through probabilistic models that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the layman.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.

I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This is essentially derived by assuming that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the layman.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.

I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This leads us to believe through probabilistic models that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the layman.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.

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Tim Ratigan
Tim Ratigan
  • 7.4k
  • 23
  • 33

I'm surprised that the other answers have not touched on the distribution of prime numbers. It is well-known that $\pi(n)$, or the number of primes $p\le n$, is asymptotic to $\int_2^n\frac1{\log x}\text dx\approx\frac{n}{\log n}$, which turns out to be a decent approximation. This is essentially derived by assuming that primes are randomly distributed across the natural numbers. This heuristic tends to be extremely accurate when describing properties of primes, including very good predictions on how many twin primes there should be before a given $n$. Terence Tao has a lot of literature on this directed toward the layman.

In short, primes behave pseudorandomly and this makes them difficult to work with in a rigorous sense.