Prime number gap

A prime gap is the difference between two consecutive primes : . The smallest prime number gap is . All other prime number gaps are even , since 2 is the only even prime number and thus the difference is formed from two odd numbers. ${\displaystyle g_{n}=p_{n+1}-p_{n}}$ ${\displaystyle g_{1}=3-2=1}$

Note: Some authors use the prime number gap to denote the number of composite numbers between two prime numbers, i.e. H. one less than the definition used here.

Occurrence of prime number gaps

• Since a gap of length 1 can only appear between an even and an odd prime number, it is obvious that it exists only once. (2 is the only even prime number).
• Whether there are infinitely many prime twins , i.e. H. Gaps of length 2 are one of the great unsolved problems in mathematics .
• Apart from the gap between 2 and 3, the length of a prime number gap is always even.
• Since there are infinitely many prime numbers, the lengths of the prime number gaps form a sequence with the initial terms:

1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4, 6, 2, 10, 6, 6, 6, 2, 6, 4, 2 ... (sequence A001223 in OEIS ).

According to the definition of one has: ${\displaystyle g_{i}}$

${\displaystyle p_{n+1}=2+\sum _{i=1}^{n}g_{i}.}$

Construction of arbitrarily large prime number gaps

For any natural number it is very easy to prove the existence of a prime number gap of at least length . Namely, let be a natural number that is not prime to any of the numbers . Then the numbers are not too prime and consequently they are not prime numbers either. The largest prime number before this sequence is therefore at most equal , the smallest after that, however, at least , so that the length of this prime number gap is at least . ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle N}$ ${\displaystyle 2,3,\ldots ,n}$ ${\displaystyle N+2,N+3,\ldots ,N+n}$ ${\displaystyle N}$ ${\displaystyle N+1}$ ${\displaystyle N+n+1}$ ${\displaystyle n}$

You have various options for creating one with the required property. In terms of proof, the easiest way to choose is the faculty , i.e. in which case the considered are even divisible by. As well can be the least common multiple of the numbers from 2 to choose . ${\displaystyle N}$ ${\displaystyle N=n!}$ ${\displaystyle N+k}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle N=\operatorname {kgV} (2,\ldots ,n)}$

The smallest possible candidates are found through the Primfakultät , . If the smallest prime number is greater than , then the following applies : H. one even automatically found a gap of length . ${\displaystyle N}$ ${\displaystyle N=n\#}$ ${\displaystyle p}$ ${\displaystyle n}$ ${\displaystyle n\#=(p-1)\#}$ ${\displaystyle p-1}$

Although in the last case the selection was made as small as possible, it is not guaranteed that the gaps found are always the first gap of the required length. In this respect, although all of these methods provide equivalent evidence that there are gaps of any size, they are only of limited use when searching for the first occurrences of large gaps. ${\displaystyle N}$

Example for n = 6

Which gaps do the procedures mentioned provide in each case ? For comparison: The first gap of length 6 occurs between 23 and 29. ${\displaystyle n=6}$

Faculty

There is 6! = 720.

Since 720 is divisible by 2, it's also 720 + 2 = 722.

Since 720 is divisible by 3, it's also 720 + 3 = 723.

Since 720 is divisible by 4, it's also 720 + 4 = 724.

Since 720 is divisible by 5, it's also 720 + 5 = 725.

Since 720 is divisible by 6, it's also 720 + 6 = 726.

So a prime number gap of at least length 6 has been found between prime number candidates 721 and 727. Since 721 is divisible by 7, the gap is even larger. In fact, it is framed by the prime numbers 719 and 727 and therefore has the length 8.

Lcm (least common multiple)

Lcm (1,…, 6) = 60 applies.

Since 60 is divisible by 2, it's also 60 + 2 = 62.

Since 60 is divisible by 3, it's also 60 + 3 = 63.

Since 60 is divisible by 4, it's also 60 + 4 = 64.

Since 60 is divisible by 5, it's also 60 + 5 = 65.

Since 60 is divisible by 6, it's also 60 + 6 = 66.

So this time we found a gap of at least 6 in length between 61 and 67. Both are "random" prime numbers; H. the length of the gap is exactly 6.

Prime Faculty

It is . ${\displaystyle 6\#=2\cdot 3\cdot 5=30}$

Since 30 is divisible by 2, it's also 30 + 2 = 32.

Since 30 is divisible by 3, it's also 30 + 3 = 33.

Since 30 and 4 are divisible by 2, it's also 30 + 4 = 34.

Since 30 is divisible by 5, it's also 30 + 5 = 35.

Since 30 and 6 are divisible by 2, it's also 30 + 6 = 36.

Again, the gap found has exactly length 6, since 31 and 37 are prime numbers.

Growth of functions

The example already shows that the faculty is by far the fastest growing among the functions considered. For , the size difference between , and is even clearer. In contrast, a gap of length 14 already appears between 113 and 127, so that even the estimate by is far from being sharp. ${\displaystyle n=6}$ ${\displaystyle n=10}$ ${\displaystyle n!=3628800}$ ${\displaystyle \operatorname {kgV} (2,\ldots ,10)=2520}$ ${\displaystyle n\#=210}$ ${\displaystyle n\#}$

Upper bounds

Joseph Bertrand showed the following natural limitation of a prime number gap: The following applies to each : between and there is at least one prime number. It follows that a prime number gap, starting at , cannot be larger than itself. ${\displaystyle n>1\ }$ ${\displaystyle n\ }$ ${\displaystyle 2n\ }$ ${\displaystyle n\ }$ ${\displaystyle n\ }$

From the prime number theorem it follows that the gaps for large ones grow logarithmically on average . It also follows from the prime number theorem: for each there is a number such that ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle \epsilon >0}$ ${\displaystyle N}$

${\displaystyle g_{n}<p_{n}\epsilon }$.

for everyone and ${\displaystyle n>N}$

${\displaystyle \lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.}$

In 1930, Guido Hoheisel showed that there is a constant such that: ${\displaystyle \theta <1}$

${\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log(x)}}{\text{ as }}x\to \infty ,}$

and thus

${\displaystyle g_{n}<p_{n}^{\theta },\,}$

for big enough . According to Hoheisel, the value of could be chosen to be close to 1 and was constantly improved over time ( Hans Heilbronn , Nikolai Grigorjewitsch Tschudakow and any , Albert Ingham , Martin Huxley , János Pintz , Baker, Harman ). ${\displaystyle n}$ ${\displaystyle \theta }$ ${\displaystyle {\frac {249}{250}}}$ ${\displaystyle \theta ={\frac {3}{4}}+\varepsilon }$ ${\displaystyle \epsilon >0}$ ${\displaystyle \theta >{\frac {5}{8}}}$ ${\displaystyle \theta ={\frac {7}{12}}}$ ${\displaystyle \theta =0{,}525}$

In 2005 Daniel Goldston , János Pintz and Cem Yıldırım proved that

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0}$

what they 2007 on

${\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .}$

improved. In 2017, Yitang Zhang showed that

${\displaystyle \liminf _{n\to \infty }g_{n}<7\cdot 10^{7},}$

and that there are thus an infinite number of prime number gaps that are smaller than 70 million. That was pushed to 600 by James Maynard and to 246 by the Polymath project.

Lower bounds

In 1931 the Finn Erik Westzynthius (1901–1980) showed that the maximum prime number gap grows more than logarithmically:

${\displaystyle \limsup _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=\infty .}$

In 1938 Robert Alexander Rankin showed that there is a constant such that ${\displaystyle c}$

${\displaystyle g_{n}>{\frac {c\log n\log \log n\log \log \log \log n}{(\log \log \log n)^{2}}}}$

is satisfied for an infinite number of values . He also showed that you can use any constant (with the Euler-Mascheroni constant ) for this. János Pintz improved this in 1997 . Paul Erdös suspected that the constant could be of any size and offered a price of 10,000 dollars for the proof. In 2014, independently of each other, James Maynard on the one hand and Terence Tao and colleagues on the other hand proved the conjecture and also that ${\displaystyle n}$ ${\displaystyle c<e^{\gamma }}$ ${\displaystyle \gamma }$ ${\displaystyle c<2e^{\gamma }}$ ${\displaystyle c}$

${\displaystyle g_{n}>{\frac {\log n\log \log n\log \log \log \log n}{\log \log \log n}}}$

for infinitely many values ​​of . ${\displaystyle n}$

assumptions

Assuming the Riemann Hypothesis , Harald Cramér showed in 1936 that

${\displaystyle g_{n}=O({\sqrt {p_{n}}}\log p_{n}),}$

using the Landau symbols . Cramer suspected that

${\displaystyle g_{n}=O\left((\log p_{n})^{2}\right).}$

According to a presumption of the Dane Ludvig Oppermann (1817-1883) is

${\displaystyle g_{n}<{\sqrt {p_{n}}}.}$

From Andrica's conjecture (a tightening of Legendre's conjecture ) it follows that

${\displaystyle g_{n}<2{\sqrt {p_{n}}}+1.}$

Polignac's conjecture says that every even number appears as a prime number gap infinitely often, for this is the twin prime conjecture . According to Zhang Yitang, she is right for one . ${\displaystyle k}$ ${\displaystyle k=2}$ ${\displaystyle k<70.000.000}$

Web links

Wikibooks: Prime Numbers: Prime  Number Gaps - Learning and Teaching Materials

• Eric W. Weisstein : Prime Gaps . In: MathWorld (English).
• The Gaps Between Primes (English)
• First occurrence prime gaps by Thomas R. Nicely (English) - The reference website and current information on prime number gaps

Individual evidence

1. ^ Hoheisel, Prime number problems in analysis, session reports of the Royal Prussian Academy of Sciences, Volume 33, 1930, pp. 3-11
2. ↑ Huxley, On the difference between consecutive primes, Inv. Math., Vol. 15, 1972, pp. 164-170
3. ^ RC Baker, G. Harman, J. Pintz, The difference between consecutive primes, II, Proceedings of the London Mathematical Society, Volume 83, 2001, pp. 532-562
4. ↑ Zhang, Buondes gaps between primes, Annals of Mathematics, vol 179, 2014, pp 1121-1174
5. ↑ James Maynard, Large gaps between primes, Annals of Mathematics, Volume 183, 2016, pp. 915-922
6. ↑ Kevin Ford, Ben Green, Sergei Konyagin, Terence Tao, Large gaps between consecutive prime numbers, Ann. of Math., Volume 183, 2016, pp. 935-974