Distribution
In 1975, number theorist Don Zagier commented that primes both
| “ | grow like weeds among the natural numbers, seeming to obey no other law than that of chance exhibit stunning regularity that there are laws governing their behavior, and that they obey these laws with almost military precision. | ” |
The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the n-th prime is known.
There are arbitrarily long sequences of consecutive non-primes, as for every positive integer the consecutive integers from to (inclusive) are all composite (as is divisible by for between and ).
Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials
with coprime integers a and b take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different such polynomials with the same b have approximately the same proportions of primes.
The corresponding question for quadratic polynomials is less well-understood.
Read more about this topic: Prime Number
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