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Oppermann's conjecture is an unsolved problem in
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
on the distribution of
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), 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 ...
s.. It is closely related to but stronger than Legendre's conjecture, Andrica's conjecture, and Brocard's conjecture. It is named after Danish mathematician Ludvig Oppermann, who announced it in an unpublished lecture in March 1877.


Statement

The conjecture states that, for every integer n>1, there is at least one prime number between : n(n-1) and n^2, and at least another prime between : n^2 and n(n+1). It can also be phrased equivalently as stating that the prime-counting function must take unequal values at the endpoints of each range.. That is: : \pi(n^2-n)<\pi(n^2)<\pi(n^2+n) for every n>1 with \pi(x) being the number of prime numbers less than or equal to x. The end points of these two ranges are a
square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
between two pronic numbers, with each of the pronic numbers being twice a pair triangular number. The sum of the pair of triangular numbers is the square.


Consequences

If the conjecture is true, then the gap size would be on the order of : g_n < \sqrt.\, This also means there would be at least two primes between n^2 and (n+1)^2 (one in the range from n^2 to n(n+1) and the second in the range from n(n+1) to (n+1)^2, strengthening Legendre's conjecture that there is at least one prime in this range. Because there is at least one non-prime between any two odd primes it would also imply Brocard's conjecture that there are at least four primes between the squares of consecutive odd primes. Additionally, it would imply that the largest possible gaps between two consecutive prime numbers could be at most proportional to twice the
square root In mathematics, a square root of a number is a number such that y^2 = x; in other words, a number whose ''square'' (the result of multiplying the number by itself, or y \cdot y) is . For example, 4 and −4 are square roots of 16 because 4 ...
of the numbers, as Andrica's conjecture states. The conjecture also implies that at least one prime can be found in every quarter revolution of the Ulam spiral.


See also

*
Bertrand's postulate In number theory, Bertrand's postulate is the theorem that for any integer n > 3, there exists at least one prime number p with :n < p < 2n - 2. A less restrictive formulation is: for every n > 1, there is always at least one ...
* Firoozbakht's conjecture *
Prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, 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 p ...


References

{{DEFAULTSORT:Oppermann's Conjecture Conjectures about prime numbers Unsolved problems in number theory