Prime gap

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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 g₁ = 1, g₂ = g₃ = 2, and g₄ = 4. The sequence (g_n) of prime gaps has been extensively studied. One also writes g(p_n) for g_n.

Contents 1 Simple observations 2 Numerical results 3 Further results 4 Conjectures about gaps between primes 5 References

[edit] Simple observations

For any prime number P larger than 2, the sequence (for the notation P# see primorial, and P_n+1 denotes the smallest prime number greater than P)

P# + 2, P# + 3, ..., P# + (P_n+1-1)

is a sequence of P_n+1-2 consecutive composite integers, implying a prime gap of at least length P_n+1-1. Therefore, there exist gaps between primes which are arbitrarily large, i.e. for any prime number P, there is an integer n with g_p > 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, wheareas the number 71 primorial (the product of all prime numbers up to and including 71, or 71*67*61*59*53*47...*5*3*2) 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.

[edit] Numerical results

As of 2006 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. [1] 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. [2]

We say that g_n is a maximal gap if g_m < g_n for all m < n. As of December 2006 the largest known maximal gap has length 1370, found by Donald E. Knuth. It is the 73rd maximal gap, and it occurs after the prime 418032645936712127. [3]

The first 73 maximal gaps (n is not listed)

Number 1 to 25 # gn pn 1 1 2 2 2 3 3 4 7 4 6 23 5 8 89 6 14 113 7 18 523 8 20 887 9 22 1129 10 34 1327 11 36 9551 12 44 15683 13 52 19609 14 72 31397 15 86 155921 16 96 360653 17 112 370261 18 114 492113 19 118 1349533 20 132 1357201 21 148 2010733 22 154 4652353 23 180 17051707 24 210 20831323 25 220 47326693

Number 26 to 50 # gn pn 26 222 122164747 27 234 189695659 28 248 191912783 29 250 387096133 30 282 436273009 31 288 1294268491 32 292 1453168141 33 320 2300942549 34 336 3842610773 35 354 4302407359 36 382 10726904659 37 384 20678048297 38 394 22367084959 39 456 25056082087 40 464 42652618343 41 468 127976334671 42 474 182226896239 43 486 241160624143 44 490 297501075799 45 500 303371455241 46 514 304599508537 47 516 416608695821 48 532 461690510011 49 534 614487453523 50 540 738832927927

Number 51 to 72 # gn pn 51 582 1346294310749 52 588 1408695493609 53 602 1968188556461 54 652 2614941710599 55 674 7177162611713 56 716 13829048559701 57 766 19581334192423 58 778 42842283925351 59 804 90874329411493 60 806 171231342420521 61 906 218209405436543 62 916 1189459969825483 63 924 1686994940955803 64 1132 1693182318746371 65 1184 43841547845541059 66 1198 55350776431903243 67 1220 80873624627234849 68 1224 203986478517455989 69 1248 218034721194214273 70 1272 305405826521087869 71 1328 352521223451364323 72 1356 401429925999153707 73 1370 418032645936712127 Gap with unknown status:  ? 1442 804212830686677669

The largest known value of g_n / log(p_n) -- usually called the merit of the gap g_n -- is 1442 / log(804212830686677669) = 34.98, found by S. Herzog and T. O. e Silva. [4] As of December 2006 it is not known whether this is a maximal gap.

[edit] Further results

It follows from Bertrand's postulate that g_n<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 < εp_n for all n > N.

Hoheisel was the first to show^[1] 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

if n is sufficiently large.

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,^[2] and to θ = 3/4 + e, for any e > 0, by Čudakov.^[3]

A major improvement is due to Ingham,^[4] who showed that if

ζ(1/2 + it) = O(t^c),

for some positive constant c, then

π(x + x^θ) - π(x) ~ x^θ/log(x).

for any θ > (1 + 4c)/(2 + 4c). 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 n³ 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 n² and (n + 1)², if n is sufficiently large (Compare Legendre's conjecture). To verify this, as of 2006 still unresolved problem, a stronger result such as Cramér's conjecture would be needed.

Huxley showed that one may choose θ = 7/12.^[5]

A recent result, due to Baker, Harman and Pintz, shows that θ may be taken to be 0.525.^[6]

[edit] 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) satisfies

Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that

At the moment, the numerical evidence seems to point in this direction. See Cramér's conjecture for more details.

Andrica's conjecture states that

[edit] References

1. ^ G. Hoheisel, Primzahlprobleme in der Analysis, Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, 33, pages 3-11, (1930)
2. ^ H. A. Heilbronn, Über den Primzahlsatz von Herrn Hoheisel, Mathematische Zeitschrift, 36, pages 394-423, (1933)
3. ^ N. G. Tchudakoff, On the difference between two neighboring prime numbers, Math. Sb., 1, pages 799-814, (1936)
4. ^ Ingham, A. E. On the difference between consecutive primes, Quarterly Journal of Mathematics (Oxford Series), 8, pages 255-266, (1937)
5. ^ Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones mathematicae 15: 164-170.
6. ^ Baker, R. C., G. Harman, G. and J. Pintz (2001). "The difference between consecutive primes, II". Proceedings of the London Mathematical Society 83: 532–562.