What does the prime number theorem state?

In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N). …

How do you use the prime number theorem?

The prime number theorem provides a way to approximate the number of primes less than or equal to a given number n. This value is called π(n), where π is the “prime counting function.” For example, π(10) = 4 since there are four primes less than or equal to 10 (2, 3, 5 and 7).

How do you get prime gap?

A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n + 1)-th and the n-th prime numbers, i.e. We have g1 = 1, g2 = g3 = 2, and g4 = 4.

Are prime gaps bounded?

But what Yitang Zhang just proved is that there are infinitely many pairs of primes that differ by at most 70,000,000. In other words, that the gap between one prime and the next is bounded by 70,000,000 infinitely often—thus, the “bounded gaps” conjecture. On first glance, this might seem a miraculous phenomenon.

Who proved the prime number theorem?

The prime number theorem, that the number of primes < x is asymptotic to x/log x, was proved (independently) by Hadamard and de la Vallee Poussin in 1896. Their proof had two elements: showing that Riemann’s zeta function ;(s) has no zeros with Sc(s) = 1, and deducing the prime number theorem from this.

How do you get prime gaps?

What are all the prime numbers?

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Are there any prime numbers next to each other?

The first few twin prime pairs are: (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73), (101, 103), (107, 109), (137, 139), … OEIS: A077800.