Cramér's conjecture

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In number theory, Cramér's conjecture, formulated originally by the Swedish mathematician Harald Cramér in 1936 ^[1], 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.^[2]

Shanks conjectures asymptotic equality of record gaps, a somewhat stronger statement:^[3]

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^[4]

$\limsup_{n\to\infty}\frac{p_{n+1}-p_n}{\log p_n}=\infty.$

1 Cramér-Granville conjecture
2 See also
3 References
4 External links

[edit] Cramér-Granville conjecture

Cramér's conjecture may be too strong. Andrew Granville in 1995 proposed^[5] there is a bound $M$ for which $p_{n+1}-p_n\,<\ M\ \ (\log p_n)^2$ . Maier suggested $M=2e^{-\gamma}\approx1.1229\ldots\$ .

[edit] See also

Prime number theorem

[edit] References

^ Harald Cramér, On the order of magnitude of the difference between consecutive prime numbers, Acta Arithmetica 2 (1936), pp. 23–46.
^ David Hawkins, "The Random Sieve", Mathematics Magazine 31 (1957), pp. 1–3.
^ Daniel Shanks, "On Maximal Gaps between Successive Primes", Mathematics of Computation 18, No. 88 (1964), pp. 646–651.
^ E. Westzynthius, Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind, Commentationes Physico-Mathematicae Helingsfors, 5 (1931), pp. 1–37.
^ A. Granville, "Harald Cramér and the distribution of prime numbers", Scandinavian Actuarial J. 1 (1995), 12—28. [1]