In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.
Informally speaking, the prime number theorem states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is about 1 / ln(N), where ln(N) is the natural logarithm of N. For example, among the positive integers up to and including N = 103 about one in seven numbers is prime, whereas up to and including N = 1010 about one in 23 numbers is prime (where ln(103)= 6.90775528. and ln(1010)=23.0258509). In other words, the average gap between consecutive prime numbers among the first N integers is roughly ln(N).
Read more about Prime Number Theorem: Statement of The Theorem, History of The Asymptotic Law of Distribution of Prime Numbers and Its Proof, Proof Methodology, Proof Sketch, Prime-counting Function in Terms of The Logarithmic Integral, Elementary Proofs, Computer Verifications, Prime Number Theorem For Arithmetic Progressions, Bounds On The Prime-counting Function, Approximations For The nth Prime Number, Table of π(x), x / Ln x, and Li(x), Analogue For Irreducible Polynomials Over A Finite Field
Famous quotes containing the words prime, number and/or theorem:
“Faith in reason as a prime motor is no longer the criterion of the sound mind, any more than faith in the Bible is the criterion of righteous intention.”
—George Bernard Shaw (1856–1950)
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—Kyle D. Pruett (20th century)
“To insure the adoration of a theorem for any length of time, faith is not enough, a police force is needed as well.”
—Albert Camus (1913–1960)