Oppermann's conjecture

The conjecture states that, for every integer x > 1, there is at least one prime number between

x(x − 1) and x²,

and at least another prime between

x² and x(x + 1).

It can also be phrased equivalently as stating that the prime-counting function must take unequal values at the endpoints of each range.[3] That is:

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

with π(x) being the number of prime numbers less than or equal to x. The end points of these two ranges are a square 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.

If the conjecture is true, then the gap size would be on the order of

${\displaystyle g_{n}<{\sqrt {p_{n}}}\,}$.

This also means there would be at least two primes between x² and (x + 1)² (one in the range from x² to x(x + 1) and the second in the range from x(x + 1) to (x + 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.[1] Additionally, it would imply that the largest possible gaps between two consecutive prime numbers could be at most proportional to twice the square root 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.

Even for small values of x, the numbers of primes in the ranges given by the conjecture are much larger than 1, providing strong evidence that the conjecture is true. However, Oppermann's conjecture has not been proved as of 2015.[1]

• Bertrand's postulate
• Firoozbakht's conjecture
• Prime number theorem