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:
“I did not know I was in my prime until afterwards.”
—Mason Cooley (b. 1927)
“In many ways, life becomes simpler [for young adults]. . . . We are expected to solve only a finite number of problems within a limited range of possible solutions. . . . It’s a mental vacation compared with figuring out who we are, what we believe, what we’re going to do with our talents, how we’re going to solve the social problems of the globe . . .and what the perfect way to raise our children will be.”
—Roger Gould (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)