# Field theory (mathematics)

In number theory, Cramér's conjecture, formulated by the Swedish mathematician Harald Cramér in 1936,[1] is an estimate for the size of gaps between consecutive prime numbers: intuitively, that gaps between consecutive primes are always small, and the conjecture quantifies asymptotically just how small they can be. It states that

${\displaystyle p_{n+1}-p_{n}=O((\log p_{n})^{2}),\ }$

where pn denotes the nth prime number, O is big O notation, and "log" is the natural logarithm. While this is the statement explicitly conjectured by Cramér, his argument actually supports the stronger statement

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

and this formulation is often called Cramér's conjecture in the literature.

Neither form of Cramér's conjecture has yet been proven or disproven.

## Conditional proven results on prime gaps

Cramér also gave a conditional proof of the much weaker statement that

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

on the assumption of the Riemann hypothesis.[1]

In the other direction, E. Westzynthius proved in 1931 that prime gaps grow more than logarithmically. That is,[2]

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

## Heuristic justification

Cramér's conjecture is based on a probabilistic model (essentially a heuristic) of the primes, in which one assumes that the probability of a natural number of size x being prime is 1/log x. This is known as the Cramér model of the primes. Cramér proved that in this model, the above conjecture holds true with probability one.[1]

## Heuristics of Shanks and Cramér–Granville conjectures

Daniel Shanks conjectured asymptotic equality of record gaps, a somewhat stronger statement than Cramér's conjecture.[3]

In the random model,

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

But this constant, ${\displaystyle c}$, may not apply to all the primes, by Maier's theorem. As pointed out by Andrew Granville,[4] a refinement of Cramér's model taking into account divisibility by small primes suggests that ${\displaystyle c\geq 2e^{-\gamma }\approx 1.1229\ldots }$, Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland. where ${\displaystyle \gamma }$ is the Euler–Mascheroni constant. Physiotherapist Rave from Cobden, has hobbies and interests which includes skateboarding, commercial property for sale developers in singapore and coin collecting. May be a travel freak and in recent years made a journey to Wet Tropics of Queensland.

Thomas Nicely has calculated many large prime gaps.[5] He measures the quality of fit to Cramér's conjecture by measuring the ratio

${\displaystyle R={\frac {\log(p_{n})}{\sqrt {p_{n+1}-p_{n}}}}.}$

He writes, “For the largest known maximal gaps, ${\displaystyle R}$ has remained near 1.13.” However, ${\displaystyle 1/R^{2}}$ is still less than 1, and it does not provide support to Granville's refinement that c should be greater than 1.

## References

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