Cramer's conjecture: Difference between revisions
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'''Cramér's conjecture''' asserts that the largest gap between adjacent primes of size N should be <math>O(\log^2 N)</math>. This is compatible with [[Cramer's random model for the primes]], and specifically with the belief that the number of primes in <math>[n,n+\log n]</math> should resemble a Poisson distribution asymptotically. | '''Cramér's conjecture''' asserts that the largest gap between adjacent primes of size N should be <math>O(\log^2 N)</math>. This is compatible with [[Cramer's random model for the primes]], and specifically with the belief that the number of primes in <math>[n,n+\log n]</math> should resemble a Poisson distribution asymptotically. | ||
If this conjecture is true, one has an easy positive answer to the [[finding primes]] project in the strongest form; one simply searches an interval of the form <math>[N, N+O(\log^2 N)]</math> for primes, where N is your favourite k-digit number. | If this conjecture is true, one has an [[An efficient algorithm exists if Cramer's conjecture holds|easy positive answer]] to the [[finding primes]] project in the strongest form; one simply searches an interval of the form <math>[N, N+O(\log^2 N)]</math> for primes, where N is your favourite k-digit number. | ||
# [[wikipedia:Cramér's_conjecture|Wikipedia entry on Cramér' | # [[wikipedia:Cramér's_conjecture|Wikipedia entry on Cramér's conjecture]] | ||
Latest revision as of 17:10, 19 August 2009
Cramér's conjecture asserts that the largest gap between adjacent primes of size N should be [math]\displaystyle{ O(\log^2 N) }[/math]. This is compatible with Cramer's random model for the primes, and specifically with the belief that the number of primes in [math]\displaystyle{ [n,n+\log n] }[/math] should resemble a Poisson distribution asymptotically.
If this conjecture is true, one has an easy positive answer to the finding primes project in the strongest form; one simply searches an interval of the form [math]\displaystyle{ [N, N+O(\log^2 N)] }[/math] for primes, where N is your favourite k-digit number.
