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In mathematics, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1937, states that
- <math>\limsup_{n\rightarrow\infty} \frac{p_{n+1}-p_n}{(\log p_n)^2} = 1<math>
where pn denotes the n-th prime number and "log" is the natural logarithm. This conjecture is unproven until today and is unlikely to be ever proved. It is based on a probabilistic model of the primes, in which one assumes that the probability that a natural number x is prime is 1/log x, and it can be proved that the above conjecture holds true with probability one. In other words, if primes are in some sense 'random' then they will likely follow the random model. Cramér also formulated another conjecture concerning prime numbers, stating that
- <math>p_{n+1}-p_n = \mathcal{O}(\sqrt{p_n}\,\log p_n)<math>
which he proved using the (as-of-yet unproven) Riemann hypothesis.fr:Conjecture de Cramér it:Congettura di Cramér zh:克拉梅爾猜想