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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.
The theorem extends Euclid's theorem that there are infinitely many prime numbers. Stronger forms of Dirichlet's theorem state that for any such arithmetic progression, the sum of the reciprocals of the prime numbers in the progression diverges and that different such arithmetic progressions with the same modulus have approximately the same ...
In number theory, primes in arithmetic progression are any sequence of at least three prime numbers that are consecutive terms in an arithmetic progression. An example is the sequence of primes (3, 7, 11), which is given by a n = 3 + 4 n {\displaystyle a_{n}=3+4n} for 0 ≤ n ≤ 2 {\displaystyle 0\leq n\leq 2} .
In number theory, the first Hardy–Littlewood conjecture [1] states the asymptotic formula for the number of prime k-tuples less than a given magnitude by generalizing the prime number theorem. It was first proposed by G. H. Hardy and John Edensor Littlewood in 1923. [2]
Prime number theorem; Proth's theorem; R. Rosser's theorem; S. Siegel–Walfisz theorem; Divergence of the sum of the reciprocals of the primes; V. Vantieghems theorem;
In number theory, the Green–Tao theorem, proved by Ben Green and Terence Tao in 2004, states that the sequence of prime numbers contains arbitrarily long arithmetic progressions. In other words, for every natural number k {\displaystyle k} , there exist arithmetic progressions of primes with k {\displaystyle k} terms.
Let (), the prime-counting function, denote the number of primes less than or equal to . If q {\displaystyle q} is a positive integer and a {\displaystyle a} is coprime to q {\displaystyle q} , we let π ( x ; q , a ) {\displaystyle \pi (x;q,a)} denote the number of primes less than or equal to x {\displaystyle x} which are equal to a ...
A general theorem of Matiyasevich says that if a set is defined by a system of Diophantine equations, it can also be defined by a system of Diophantine equations in only 9 variables. [8] Hence, there is a prime-generating polynomial inequality as above with only 10 variables. However, its degree is large (in the order of 10 45). On the other ...