A prime number (or a prime) is a natural number that has exactly two distinct natural number divisors: 1 and itself. The smallest twenty-five prime numbers (all the prime numbers under 100) are:
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
Despite being intensely studied, there remain some open questions around prime numbers which can be stated simply.There is no known useful formula that yields all of the prime numbers and no composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large can be modeled. The first result in that direction is the prime number theorem which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or the logarithm of n. This statement has been proven since the end of the 19th century. The unproven Riemann hypothesis dating from 1859 implies a refined statement concerning the distribution of primes.
Despite being intensely studied, there remain some open questions around prime numbers which can be stated simply.There is no known useful formula that yields all of the prime numbers and no composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large can be modeled. The first result in that direction is the prime number theorem which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or the logarithm of n. This statement has been proven since the end of the 19th century. The unproven Riemann hypothesis dating from 1859 implies a refined statement concerning the distribution of primes.
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of January 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize (sometimes called a Millennium Prize) being awarded by the institute. Only the Poincaré conjecture has been solved, by Grigori Perelman, who declined the award.
The seven problems are:
- P versus NP problem
- Hodge conjecture
- Poincaré conjecture (solved, see solution of the Poincaré conjecture)
- Riemann hypothesis
- Yang–Mills existence and mass gap
- Navier–Stokes existence and smoothness
- Birch and Swinnerton-Dyer conjecture
Riemann hypothesis
In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about the distribution of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesi s for curves over finite fields.
Δεν υπάρχουν σχόλια:
Δημοσίευση σχολίου