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A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.
A twin prime is a prime number that is either 2 less or 2 more than another prime number—for example, either member of the twin prime pair or (41, 43). In other words, a twin prime is a prime that has a prime gap of two. Sometimes the term twin prime is used for a pair of twin primes; an alternative name for this is prime twin or prime pair.
19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.
In number theory, a sphenic number is a positive integer that is the product of three distinct prime numbers. Because there are infinitely many prime numbers, there are also infinitely many sphenic numbers.
In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936, is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they must be. It states that
In number theory, sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because both are prime and 11 − 5 = 6.
23 (twenty-three) is the natural number following 22 and preceding 24.
800 is the natural number following 799 and preceding 801.
In number theory, a Smith number is a composite number for which, in a given number base, the sum of its digits is equal to the sum of the digits in its prime factorization in the same base. In the case of numbers that are not square-free, the factorization is written without exponents, writing the repeated factor as many times as needed.
In number theory, a prime quadruplet is a set of four prime numbers of the form {p, p + 2, p + 6, p + 8}. This represents the closest possible grouping of four primes larger than 3, and is the only prime constellation of length 4.
In number theory, Polignac's conjecture was made by Alphonse de Polignac in 1849 and states:
In number theory, a balanced prime is a prime number with equal-sized prime gaps above and below it, so that it is equal to the arithmetic mean of the nearest primes above and below. Or to put it algebraically, the th prime number is a balanced prime if
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)-st and the n-th prime numbers, i.e.
At the 1912 International Congress of Mathematicians, Edmund Landau listed four basic problems about prime numbers. These problems were characterised in his speech as "unattackable at the present state of mathematics" and are now known as Landau's problems. They are as follows:
- Goldbach's conjecture: Can every even integer greater than 2 be written as the sum of two primes?
- Twin prime conjecture: Are there infinitely many primes p such that p + 2 is prime?
- Legendre's conjecture: Does there always exist at least one prime between consecutive perfect squares?
- Are there infinitely many primes p such that p − 1 is a perfect square? In other words: Are there infinitely many primes of the form n2 + 1?
In number theory, a prime k-tuple is a finite collection of values representing a repeatable pattern of differences between prime numbers. For a k-tuple (a, b, …), the positions where the k-tuple matches a pattern in the prime numbers are given by the set of integers n such that all of the values (n + a, n + b, …) are prime. Typically the first value in the k-tuple is 0 and the rest are distinct positive even numbers.
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, which is given by for .
János Pintz is a Hungarian mathematician working in analytic number theory. He is a fellow of the Rényi Mathematical Institute and is also a member of the Hungarian Academy of Sciences. In 2014, he received the Cole Prize.
YitangZhang is a Chinese-American mathematician primarily working on number theory and a professor of mathematics at the University of California, Santa Barbara since 2015.
James Alexander Maynard is an English mathematician working in analytic number theory and in particular the theory of prime numbers. In 2017, he was appointed Research Professor at Oxford. Maynard is a fellow of St John's College, Oxford. He was awarded the Fields Medal in 2022 and the New Horizons in Mathematics Prize in 2023.
In number theory, a cluster prime is a prime number p such that every even positive integer k ≤ p − 3 can be written as the difference between two prime numbers not exceeding p. For example, the number 23 is a cluster prime because 23 − 3 = 20, and every even integer from 2 to 20, inclusive, is the difference of at least one pair of prime numbers not exceeding 23:
References
- 1 2 Weisstein, Eric W. "Interprime". mathworld.wolfram.com. Retrieved 2020-08-10.
- ↑ Caldwell, Chris K. "The Top Twenty: Twin Primes". The Prime Pages. Retrieved 27 February 2017.
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