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.

Categories: Analytic number theory | Conjectures

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07-14-2008 23:18:10
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