Cramér's conjecture

In mathematics, Cramér's conjecture, formulated by Swedish mathematician Harald Cramér in 1937, states that

$\limsup _{n\rightarrow \infty }{\frac {p_{n+1}-p_{n}}{(\ln p_{n})^{2}}}=1$

where p_n denotes the n-th prime number; this conjecture is unproven until today. Cramér also formulated another conjecture concerning prime numbers, stating that

$p_{n+1}-p_{n}={\mathcal {O}}({\sqrt {p_{n}}}\,\ln p_{n})$

which he proved using the (as-of-yet unproven) Riemann hypothesis.