section_path: Elementary properties/Open questions
section_name: Open questions
paragraph_text: Open questions
  
  Many conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n {\displaystyle n} greater than 2 can be written as a sum of two primes. As of 2014, this conjecture has been verified for all numbers up to n = 4 ⋅ 10 18. {\displaystyle n=4\cdot 10^{18}.} Weaker statements than this have been proven, for example, Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime (the product of two primes). Also, any even integer greater than 10 can be written as the sum of six primes. The branch of number theory studying such questions is called additive number theory.
  
  Another type of problem concerns prime gaps, the differences between consecutive primes.The existence of arbitrarily large prime gaps can be seen by noting that the sequence n ! + 2, n ! + 3, …, n ! + n {\displaystyle n!+2,n!+3,\dots,n!+n} consists of n − 1 {\displaystyle n-1} composite numbers, for any natural number n. {\displaystyle n.} However, large prime gaps occur much earlier than this argument shows. For example, the first prime gap of length 8 is between the primes 89 and 97, much smaller than 8 ! = 40320. {\displaystyle 8!=40320.} It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2; this is the twin prime conjecture. Polignac's conjecture states more generally that for every positive integer k, {\displaystyle k,} there are infinitely many pairs of consecutive primes that differ by 2 k. {\displaystyle 2k.} Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, and Oppermann's conjecture all suggest that the largest gaps between primes from 1 {\displaystyle 1} to n {\displaystyle n} should be at most approximately n, {\displaystyle {\sqrt {n}},} a result that is known to follow from the Riemann hypothesis, while the much stronger Cramér conjecture sets the largest gap size at O ((log n) 2). {\displaystyle O((\log n)^{2}).} Prime gaps can be generalized to prime k {\displaystyle k} -tuples, patterns in the differences among more than two prime numbers. Their infinitude and density are the subject of the first Hardy–Littlewood conjecture, which can be motivated by the heuristic that the prime numbers behave similarly to a random sequence of numbers with density given by the prime number theorem.