Cramér's conjecture

In number theory, Cramér's conjecture, formulated originally by the Swedish mathematician Harald Cramér in 1936, states that

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

where p_n denotes the nth prime number and "log" is the natural logarithm. This conjecture has not been proven or disproven, and is unlikely to be proven in the foreseeable future. It is based on a probabilistic model (essentially a heuristic) of the primes, in which one assumes that the probability that a natural number x is prime is 1/log x. This is known as the Cramér model of the primes. From this it can be proved that the above conjecture holds true with probability one.

Shanks conjectures asymptotic equality of record gaps, a somewhat stronger statement:

Cramér also formulated another conjecture concerning prime gaps, stating that

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

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

In addition, E. Westzynthius proved the following in 1931

$\limsup _{n\to \infty }{\frac {p_{n+1}-p_{n}}{\log p_{n}}}=\infty .$

Cramér-Granville conjecture

Cramér's conjecture may be too strong. Andrew Granville in 1995 proposed there is a bound $M$ for which $p_{n+1}-p_{n}<M(\log {p_{n}})^{2}$. Maier suggested $M=2e^{-\gamma }\approx 1.1229\ldots .\$

Nicely has calculated many large prime gaps. He measures the quality of fit to Cramér's conjecture by measuring the ratio R of the logarithm of a prime to the square root of the gap; he writes, “For the largest known maximal gaps, R has remained near 1.13,” showing that, at least within the range of his calculation, the Granville refinement of Cramér's conjecture seems to be a good fit to the data.

See also

• Prime number theorem

External links

• Weisstein, Eric W. "Cramér Conjecture". MathWorld.
• Weisstein, Eric W. "Cramér-Granville Conjecture". MathWorld.