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Prime number theorem
THEOREM IN NUMBER THEORY
Prime Number Theorem; Dusart's inequality; Distribution of prime numbers; The Prime Number Theorem; Distribution of primes; Prime number theorem for arithmetic progressions; Prime number race
In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs.
prime number theorem
THEOREM IN NUMBER THEORY
Prime Number Theorem; Dusart's inequality; Distribution of prime numbers; The Prime Number Theorem; Distribution of primes; Prime number theorem for arithmetic progressions; Prime number race
<mathematics> The number of prime numbers less than x is
about x/log(x). Here "is about" means that the ratio of the
two things tends to 1 as x tends to infinity. This was first
conjectured by Gauss in the early 19th century, and was
proved (independently) by Hadamard and de la Vall'ee Poussin
in 1896. Their proofs relied on complex analysis, but Erdos
and Selberg later found an "elementary" proof.
(1995-04-10)
prime
![The [[Gaussian prime]]s with norm less than 500](https://commons.wikimedia.org/wiki/Special:FilePath/Gaussian primes.png?width=200 "The [[Gaussian prime]]s with norm less than 500")
![relative error]] of <math>\tfrac{n}{\log n}</math> and the logarithmic integral <math>\operatorname{Li}(n)</math> as approximations to the [[prime-counting function]]. Both relative errors decrease to zero as <math>n</math> grows, but the convergence to zero is much more rapid for the logarithmic integral.](https://commons.wikimedia.org/wiki/Special:FilePath/Prime-counting relative error.svg?width=200 "relative error]] of <math>\tfrac{n}{\log n}</math> and the logarithmic integral <math>\operatorname{Li}(n)</math> as approximations to the [[prime-counting function]]. Both relative errors decrease to zero as <math>n</math> grows, but the convergence to zero is much more rapid for the logarithmic integral.")
POSITIVE INTEGER WITH EXACTLY TWO DIVISORS, 1 AND ITSELF
Primes; Prime numbers; Prime factor; Primality; Prime Numbers; Prime factors; Odd prime; 1 no longer prime; Prime divisor; Prime numbers in nature; Even primes; Prime Number; Infinity of primes; Prime-Numbers; Euclidean prime number theorem; Table Of Primes List; Prime; Primalities; Prime-number; Uncompound number; Odd prime number; Ω(n); Primality of 1; A000040; 1 is not a prime number; Prime (number); Primenumber; Primality of one; Infinity of the primes; Draft:The first mathematical of the prime numbers; Draft:Integer X prime matrix; Prime (mathematics)
I. a.
1.
First, original, primitive, primal, primeval, primordial, pristine, aboriginal.
2.
Highest, chief, principal.
3.
Early, blooming.
4.
First-rate, capital, excellent.
II. n.
1.
Beginning, opening, first part, earliest stage, dawn, morning.
2.
Youth, spring of life, early days.
3.
Perfection, flower, greatest beauty, health, or strength, best days, cream.
