Prime gap

A prime gap is the difference between two successive prime numbers. The "n"-th prime gap, denoted "g"_"n", is the difference between the ("n"+1)-th and the "n"-th prime number, i.e.

: "g"_"n" = "p"_{"n" + 1} − "p"_"n".

We have "g"₁ = 1, "g"₂ = "g"₃ = 2, and "g"₄ = 4. The sequence ("g"_"n") of prime gaps has been extensively studied. One also writes "g"("p"_"n") for "g"_"n".

The first 30 prime gaps are:

: 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 OEIS|id=A001223

imple observations

For any prime number "P", we write "P"# for "P primorial", that is, the the product of all prime numbers up to and including "P". If "Q" is the prime number following "P", then the sequence

: "P"# + 2, "P"# + 3, ..., "P"# + (Q-1)

is a sequence of "Q"-2 consecutive composite integers, so here there is a prime gap of at least length "Q"-1. Therefore, there exist gaps between primes which are arbitrarily large, i.e. for any prime number "P", there is an integer "n" with "g"_"n" > "P" (This is seen by choosing "n" so that "p"_"n" is the greatest prime number less than "P"# + 2).

In reality, prime gaps of "n" numbers can occur at numbers much smaller than "n"#. For instance, the smallest sequence of 71 consecutive composite numbers occurs between 31398 and 31468, whereas 71# has "twenty-seven digits" - its full decimal expansion being 557940830126698960967415390.

Although the average gap between primes increases as the natural logarithm of the integer, the ratio of the maximum prime gap to the integers involved also increases as larger and larger numbers and gaps are encountered.

In the opposite direction, the twin prime conjecture asserts that "g"_"n" = 2 for infinitely many integers "n".

Numerical results

As of 2007 the largest known prime gap with identified probable prime gap ends has length 2254930, with 86853-digit probable primes found by H. Rosenthal and J. K. Andersen. [http://hjem.get2net.dk/jka/math/primegaps/megagap2.htm] The largest known prime gap with identified proven primes as gap ends has length 337446, with 7996-digit primes found by T. Alm, J. K. Andersen and François Morain. [http://hjem.get2net.dk/jka/math/primegaps/gap337446.htm]

We say that "g"_"n" is a "maximal gap" if "g"_"m" < "g"_"n" for all "m" < "n".
As of April 2007 the largest known maximal gap has length 1442, found by Siegfried Herzog and Tomás Oliveira e Silva. It is the 74th maximal gap, and it occurs after the prime 804212830686677669. [http://hjem.get2net.dk/jka/math/primegaps/maximal.htm]

The largest known value of (digit length of "g"_"n") / ln("p"_"n") -- usually called the "merit" of the gap "g"_"n" -- is 1442 / ln(804212830686677669) = 34.98. [http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm#top20merit]

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Further results

Upper bounds

Bertrand's postulate states that there is always a prime number between "k" and 2"k", so in particular "p"_"n"+1 &lt; 2"p"_"n", which means "g"_"n" &lt; "p"_"n".

The prime number theorem says that the "average length" of the gap between a prime "p" and the next prime is ln "p". Of course, the actual length of the gap might be much more or less than this. However, from the prime number theorem one can also easily deduce an upper bound on the length of prime gaps: for every ε > 0, there is a number "N" such that "g"_"n" < ε"p"_"n" for all "n" > "N".

Hoheisel was the first to showG. Hoheisel, "Primzahlprobleme in der Analysis", Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 33, pages 3-11, (1930)] that there exists a constant θ < 1 such that

:π("x" + "x"^θ) - π("x") ~ "x"^θ/log("x"), as "x" tends to infinity,

hence showing that

:

for sufficiently large "n".

One deduces that the gaps get arbitrarily small in proportion to the primes: the quotient "g"_"n"/"p"_"n" approaches zero as "n" goes to infinity.

Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by Heilbronn,H. A. Heilbronn, "Über den Primzahlsatz von Herrn Hoheisel", Mathematische Zeitschrift, 36, pages 394-423, (1933)] and to θ = 3/4 + ε, for any ε > 0, by Čudakov.N. G. Tchudakoff, "On the difference between two neighboring prime numbers", Math. Sb., 1, pages 799-814, (1936)]

A major improvement is due to Ingham,Ingham, A. E. "On the difference between consecutive primes", Quarterly Journal of Mathematics (Oxford Series), 8, pages 255-266, (1937)] who showed that if

:ζ(1/2 + i"t") = O("t"^"c"),

for some positive constant "c", then

:π("x" + "x"^θ) - п(x) ~ "x"^θ/log(x).

for any θ > (1 + 4"c")/(2 + 4"c"). Here, as usual, ζ denotes the Riemann zeta function and π the prime-counting function. Knowing that any "c" > 1/6 is admissible, one obtains that θ may be any number greater than 5/8.

An immediate consequence of Ingham's result is that there is always a prime number between "n"³ and ("n" + 1)³ if "n" is sufficiently large. Note however that not even the Lindelöf hypothesis, which assumes that we can take "c" to be any positive number, implies that there is a prime number between "n"² and ("n" + 1)², if "n" is sufficiently large (see Legendre's conjecture). To verify this, a stronger result such as Cramér's conjecture would be needed.

Huxley showed that one may choose θ = 7/12.cite journal | last = Huxley | first = M. N. | year = 1972 | title = "On the Difference between Consecutive Primes" | journal = Inventiones mathematicae | volume = 15 | pages = 164–170 | doi = 10.1007/BF01418933]

A recent result, due to Baker, Harman and Pintz, shows that θ may be taken to be 0.525.cite journal | last = Baker | first = R. C. | coauthors = G. Harman, G. and J. Pintz | year = 2001 | title = "The difference between consecutive primes, II" | journal = Proceedings of the London Mathematical Society | volume = 83 | pages = 532–562 | doi = 10.1112/plms/83.3.532]

Lower bounds

Robert Rankin proved the existence of a constant "c" &gt; 0 such that the inequality:$g\_n\; >\; frac\{clog\; nloglog\; nloglogloglog\; n\}\{(logloglog\; n)^2\}$holds for infinitely many values "n". The best known value of the constant "c" is currently "c" = 2"e"^γ, where γ is the Euler-Mascheroni constant. [J. Pintz, "Very large gaps between consecutive primes", J. Number Theory, 63, pages 286&ndash;301, (1997).] Paul Erdős offered a $5,000 prize for a proof or disproof that the constant "c" in the above inequality may be taken arbitrarily large.R.K. Guy, "Unsolved problems in number theory, Third edition", Springer, (2004), p.31.]

Conjectures about gaps between primes

Even better results are possible if it is assumed that the Riemann hypothesis is true. Harald Cramér proved that, under this assumption, the gap "g"("p"_"n") satisfies:$g(p\_n)\; =\; O(sqrt\{p\_n\}\; ln\; p\_n).$Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that:$g(p\_n)\; =\; Oleft((ln\; p\_n)^2\; ight).$At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.

Andrica's conjecture states that:This is a slight strengthening of Legendre's conjecture that between successive square numbers there is always a prime.

As an arithmetic function

The gap "g"_"n" between the "n"th and ("n" + 1)st prime numbers is an example of an arithmetic function. In this context it is usually denoted "d"_"n" and called the prime difference function. The function is neither multiplicative nor additive.

References

External links

* Thomas R. Nicely, [http://www.trnicely.net/ Some Results of Computational Research in Prime Numbers -- Computational Number Theory] . This reference web site includes a list of all first known occurrence prime gaps.
*MathWorld|urlname=PrimeDifferenceFunction|title=Prime Difference Function
*planetmath reference|id=3143|title=Prime Difference Function