In mathematics, Oppermann's conjecture, named after L. Oppermann, relates to the distribution of the prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

s. It states that, for any integer x > 1, there is at least one prime between

x(x − 1) and x²,

and at least another prime between

x² and x(x + 1).

Alternative statement

Let π be the prime-counting function:

π(x) = the number of prime numbers less than or equal to x.

Then

π(x² − x) < π(x²) < π(x² + x) for x > 1.

This means that between the square of a number x and the square of the same number plus (or minus) that number, there is a prime number

Prime number

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

.

If true, this would entail the unproven Legendre conjecture and Andrica conjecture. Oppermann's has not been proved as of December 2010.

See also

• Bertrand's postulate
• Brocard's conjecture
• Prime gap
• Prime number theorem
  Prime number theorem
  In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....