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| A '''prime gap''' is the difference between two successive [[prime number]]s. The ''n''-th prime gap, denoted ''g''<sub>''n''</sub> or ''g''(''p''<sub>''n''</sub>) is the difference between the (''n'' + 1)-th and the
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| ''n''-th prime numbers, i.e.
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| :<math>g_n = p_{n + 1} - p_n.\ </math>
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| We have ''g''<sub>1</sub> = 1, ''g''<sub>2</sub> = ''g''<sub>3</sub> = 2, and ''g''<sub>4</sub> = 4. The sequence (''g''<sub>''n''</sub>) of prime gaps has been extensively studied, however many questions and conjectures remain unanswered.
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| The first 30 prime gaps are:
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| : 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 {{OEIS2C|id=A001223}}.
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| ==Simple observations==
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| The first, smallest, and only odd prime gap is 1 between the only even prime number, 2, and the first odd prime, 3. All other prime gaps are even.
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| For any prime number ''P'', we write ''P''# for ''P [[primorial]]'', that is, the [[product (mathematics)|product]] of all prime numbers up to and including ''P''. If ''Q'' is the prime number following ''P'', then the sequence
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| : <math>P\#+2, P\#+3,\ldots,P\#+(Q-1)</math>
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| 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''<sub>''n''</sub> ≥ ''P''. (This is seen by choosing ''n'' so that ''p''<sub>''n''</sub> is the greatest prime number less than ''P''# + 2.) Another way to see that arbitrarily large prime gaps must exist is the fact that the density of primes approaches zero, according to the [[prime number theorem]]. In fact, by this theorem, ''P''# is very roughly a number the size of exp(''P''), and near exp(''P'') the ''average'' distance between consecutive primes is ''P''.
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| In reality, prime gaps of ''P'' numbers can occur at numbers much smaller than ''P''#. 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.
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| 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.
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| In the opposite direction, the [[twin prime conjecture]] asserts that {{nowrap|1=''g''<sub>''n''</sub> = 2}} for infinitely many integers ''n''.
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| ==Numerical results==
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| {{As of|2012}} 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.<ref>[http://users.cybercity.dk/~dsl522332/math/primegaps/megagap2.htm Largest known prime gap<!-- Bot generated title -->]</ref> 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]].<ref>[http://users.cybercity.dk/~dsl522332/math/primegaps/gap337446.htm A proven prime gap of 337446<!-- Bot generated title -->]</ref>
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| We say that ''g''<sub>''n''</sub> is a ''maximal gap'' if ''g''<sub>''m''</sub> < ''g''<sub>''n''</sub> for all ''m'' < ''n''.
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| {{As of|2009|8}} the largest known maximal gap has length 1476, found by Tomás Oliveira e Silva. It is the 75th maximal gap, and it occurs after the prime 1425172824437699411.<ref>[http://users.cybercity.dk/~dsl522332/math/primegaps/maximal.htm Maximal Prime Gaps<!-- Bot generated title -->]</ref> Other record maximal gap terms can be found at {{OEIS2C|id=A002386}}.
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| Usually the [[ratio]] of ''g''<sub>''n''</sub> / ln(''p''<sub>''n''</sub>) is called the ''merit'' of the gap ''g''<sub>''n''</sub> . As of January 2012, the largest known merit value is 66520 / ln(1931*1933#/7230 - 30244) ≈ 35.4244594 where 1933# indicates the primorial of 1933. This number, 1931*1933#/7230 - 30244, is a 816-digit prime. The next largest known merit value is 1476 / ln(1425172824437699411) ≈ 35.31.<ref>[http://users.cybercity.dk/~dsl522332/math/primegaps/gaps20.htm#top20merit The Top-20 Prime Gaps<!-- Bot generated title -->]</ref><ref name="trnicely.net">[http://www.trnicely.net/#MaxMerit NEW PRIME GAP OF MAXIMUM KNOWN MERIT]</ref> Other record merit terms can be found at {{OEIS2C|id=A111870}}.
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| The Cramer-Shanks-Granville ratio is the ratio of ''g''<sub>''n''</sub> / (ln(''p''<sub>''n''</sub>))^2.<ref name="trnicely.net"/> The greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at {{OEIS2C|id=A111943}}.
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| [[File:Wikipedia primegaps.png|thumb|350px|Prime gap function]]
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| {| border="0" cellpadding="0" cellspacing="0"
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| |+ '''The first 75 maximal gaps''' (''n'' is not listed)
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| {| class="wikitable" style="text-align:right"
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| |+ Number 1 to 25
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| |-
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| ! # !! ''g<sub>n</sub>'' !! ''p<sub>n</sub>''
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| |-
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| | 1|| 1 || 2
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| |-
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| | 2|| 2 || 3
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| |-
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| | 3|| 4 || 7
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| |-
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| | 4|| 6 || 23
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| |-
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| | 5|| 8 || 89
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| |-
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| | 6|| 14 || 113
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| |-
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| | 7|| 18 || 523
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| |-
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| | 8|| 20 || 887
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| |-
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| | 9|| 22 || 1,129
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| |-
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| |10|| 34 || 1,327
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| |-
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| |11|| 36 || 9,551
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| |-
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| |12|| 44 || 15,683
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| |-
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| |13|| 52 || 19,609
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| |-
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| |14|| 72 || 31,397
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| |-
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| |15|| 86 || 155,921
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| |-
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| |16|| 96 || 360,653
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| |-
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| |17|| 112 || 370,261
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| |-
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| |18|| 114 || 492,113
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| |-
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| |19|| 118 || 1,349,533
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| |-
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| |20|| 132 || 1,357,201
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| |-
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| |21|| 148 || 2,010,733
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| |-
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| |22|| 154 || 4,652,353
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| |-
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| |23|| 180 || 17,051,707
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| |-
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| |24|| 210 || 20,831,323
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| |-
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| |25|| 220 || 47,326,693
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| |}
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| {| class="wikitable" style="text-align:right"
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| |+ Number 26 to 50
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| |-
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| ! # !! ''g<sub>n</sub>'' !! ''p<sub>n</sub>''
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| |-
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| |26|| 222 || 122,164,747
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| |-
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| |27|| 234 || 189,695,659
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| |-
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| |28|| 248 || 191,912,783
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| |-
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| |29|| 250 || 387,096,133
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| |-
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| |30|| 282 || 436,273,009
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| |-
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| |31|| 288 || 1,294,268,491
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| |-
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| |32|| 292 || 1,453,168,141
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| |-
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| |33|| 320 || 2,300,942,549
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| |-
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| |34|| 336 || 3,842,610,773
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| |-
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| |35|| 354 || 4,302,407,359
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| |-
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| |36|| 382 || 10,726,904,659
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| |-
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| |37|| 384 || 20,678,048,297
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| |-
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| |38|| 394 || 22,367,084,959
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| |-
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| |39|| 456 || 25,056,082,087
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| |-
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| |40|| 464 || 42,652,618,343
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| |-
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| |41|| 468 || 127,976,334,671
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| |-
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| |42|| 474 || 182,226,896,239
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| |-
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| |43|| 486 || 241,160,624,143
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| |-
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| |44|| 490 || 297,501,075,799
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| |-
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| |45|| 500 || 303,371,455,241
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| |-
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| |46|| 514 || 304,599,508,537
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| |-
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| |47|| 516 || 416,608,695,821
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| |-
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| |48|| 532 || 461,690,510,011
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| |-
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| |49|| 534 || 614,487,453,523
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| |-
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| |50|| 540 || 738,832,927,927
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| |}
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| {| class="wikitable" style="text-align:right"
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| |+ Number 51 to 75
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| |-
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| ! # !! ''g<sub>n</sub>'' !! ''p<sub>n</sub>''
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| |-
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| |51|| 582 || 1,346,294,310,749
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| |-
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| |52|| 588 || 1,408,695,493,609
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| |-
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| |53|| 602 || 1,968,188,556,461
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| |-
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| |54|| 652 || 2,614,941,710,599
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| |-
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| |55|| 674 || 7,177,162,611,713
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| |-
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| |56|| 716 || 13,829,048,559,701
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| |-
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| |57|| 766 || 19,581,334,192,423
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| |-
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| |58|| 778 || 42,842,283,925,351
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| |-
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| |59|| 804 || 90,874,329,411,493
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| |-
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| |60|| 806 || 171,231,342,420,521
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| |-
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| |61|| 906 || 218,209,405,436,543
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| |-
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| |62|| 916 || 1,189,459,969,825,483
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| |-
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| |63|| 924 || 1,686,994,940,955,803
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| |-
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| |64||1,132 || 1,693,182,318,746,371
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| |-
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| |65||1,184 || 43,841,547,845,541,059
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| |-
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| |66||1,198 || 55,350,776,431,903,243
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| |-
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| |67||1,220 || 80,873,624,627,234,849
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| |-
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| |68||1,224 || 203,986,478,517,455,989
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| |-
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| |69||1,248 || 218,034,721,194,214,273
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| |-
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| |70||1,272 || 305,405,826,521,087,869
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| |-
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| |71||1,328 || 352,521,223,451,364,323
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| |-
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| |72||1,356 || 401,429,925,999,153,707
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| |-
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| |73||1,370 || 418,032,645,936,712,127
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| |-
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| |74||1,442 || 804,212,830,686,677,669
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| |-
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| |75||1,476 || 1,425,172,824,437,699,411
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| |}
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| |}
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| ==Further results==
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| ===Upper bounds===
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| [[Bertrand's postulate]] states that there is always a prime number between ''k'' and 2''k'', so in particular ''p''<sub>''n''+1</sub> < 2''p''<sub>''n''</sub>, which means ''g''<sub>''n''</sub> < ''p''<sub>''n''</sub>.
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| The [[prime number theorem]] says that the "average length" of the gap between a prime ''p'' and the next prime is ln ''p''. The actual length of the gap might be much more or less than this. However, from the prime number theorem one can also deduce an upper bound on the length of prime gaps: for every ε > 0, there is a number ''N'' such that ''g''<sub>''n''</sub> < ε''p''<sub>''n''</sub> for all ''n'' > ''N''.
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| One can deduce that the gaps get arbitrarily smaller in proportion to the primes: the quotient ''g''<sub>''n''</sub>/''p''<sub>''n''</sub> [[limit (mathematics)|approaches]] zero as ''n'' goes to infinity.
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| [[Guido Hoheisel|Hoheisel]] was the first to show<ref name="Hoheisel">{{cite journal |first=G. |last=Hoheisel |title=Primzahlprobleme in der Analysis |journal=Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin |volume=33 |issue= |pages=3–11 |year=1930 }}</ref> that there exists a constant θ < 1 such that
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| :<math>\pi(x + x^\theta) - \pi(x) \sim \frac{x^\theta}{\log(x)}\text{ as }x\text{ tends to infinity,}</math>
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| hence showing that
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| :<math>g_n<p_n^\theta,\,</math>
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| for [[sufficiently large]] ''n''.
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| Hoheisel obtained the possible value 32999/33000 for θ. This was improved to 249/250 by [[Hans Heilbronn|Heilbronn]],<ref name="Heilbronn">{{cite journal |first=H. A. |last=Heilbronn |title=Über den Primzahlsatz von Herrn Hoheisel |journal=Mathematische Zeitschrift |volume=36 |issue=1 |pages=394–423 |year=1933 |doi=10.1007/BF01188631 }}</ref> and to θ = 3/4 + ε, for any ε > 0, by [[Nikolai Chudakov|Chudakov]].<ref name="Tchudakoff">{{cite journal |first=N. G. |last=Tchudakoff |title=On the difference between two neighboring prime numbers |journal=Math. Sb. |volume=1 |issue= |pages=799–814 |year=1936 |doi= }}</ref>
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| A major improvement is due to [[Albert Ingham|Ingham]],<ref name="Ingham">{{cite journal |last=Ingham |first=A. E. |title=On the difference between consecutive primes |journal=Quarterly Journal of Mathematics |series=Oxford Series |volume=8 |issue=1 |pages=255–266 |year=1937 |doi=10.1093/qmath/os-8.1.255 }}</ref> who showed that if
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| :<math>\zeta(1/2 + it)=O(t^c)\,</math>
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| for some positive constant ''c'', where ''O'' refers to the [[big O notation]], then
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| :<math>\pi(x + x^\theta) - \pi(x) \sim \frac{x^\theta}{\log(x)}</math>
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| 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.
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| An immediate consequence of Ingham's result is that there is always a prime number between ''n''<sup>3</sup> and (''n'' + 1)<sup>3</sup> 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''<sup>2</sup> and (''n'' + 1)<sup>2</sup>, if ''n'' is sufficiently large (see [[Legendre's conjecture]]). To verify this, a stronger result such as [[Cramér's conjecture]] would be needed.
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| [[Martin Huxley|Huxley]] showed that one may choose θ = 7/12.<ref name="huxley">{{cite journal |last=Huxley |first=M. N. |year=1972 |title=On the Difference between Consecutive Primes |journal=Inventiones Mathematicae |volume=15 |issue=2 |pages=164–170 |doi=10.1007/BF01418933 }}</ref>
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| A result, due to Baker, [[Glyn Harman|Harman]] and [[János Pintz|Pintz]] in 2001, shows that θ may be taken to be 0.525.<ref name="baker">{{cite journal |last=Baker |first=R. C. |first2=G. |last2=Harman |first3=G. |last3=Pintz |first4=J. |last4=Pintz |year=2001 |title=The difference between consecutive primes, II |journal=Proceedings of the London Mathematical Society |volume=83 |issue=3 |pages=532–562 |doi=10.1112/plms/83.3.532 }}</ref>
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| In 2005, [[Daniel Goldston]], [[János Pintz]] and [[Cem Yıldırım]] proved that
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| :<math>\liminf_{n\to\infty}\frac{g_n}{\log p_n}=0</math>
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| and later improved it<ref>{{cite web|url=http://arxiv.org/abs/0710.2728|title=Primes in Tuples II|publisher=ArXiv|accessdate=2013-11-23}}</ref> to
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| :<math>\liminf_{n\to\infty}\frac{g_n}{\sqrt{\log p_n}(\log\log p_n)^2}<\infty.</math>
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| In 2013, [[Yitang Zhang]] proved that <math>\liminf_{n\to\infty} g_n < 7\cdot 10^7</math>, meaning infinitely many gaps do not exceed 70 million.<ref>{{cite journal | url = http://annals.math.princeton.edu/articles/7954 | title = Bounded gaps between primes | first = Yitang | last = Zhang | journal = Annals of Mathematics | publisher = Princeton University and the Institute for Advanced Study | accessdate =August 16, 2013}}</ref> A [[Polymath Project]] collaborative effort to optimize Zhang’s bound managed to lower the bound to 4680 on July 20, 2013.<ref>{{cite web|url=http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|title=Bounded gaps between primes|publisher=Polymath|accessdate=2013-07-21}}</ref> As of November 28, 2013, Thomas Engelsma claims to have reduced the bound to N = 576.<ref>{{cite web|url=http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes|title=Bounded gaps between primes|publisher=Polymath|accessdate=2013-11-28}}</ref>
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| ===Lower bounds===
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| [[Robert Alexander Rankin|Robert Rankin]] proved the existence of a constant ''c'' > 0 such that the inequality
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| :<math>g_n > \frac{c\log n\log\log n\log\log\log\log n}{(\log\log\log n)^2}</math>
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| holds for infinitely many values ''n'': he showed that one can take ''c'' = ''e''<sup>γ</sup>, where γ is the [[Euler–Mascheroni constant]]. The best known value of the constant ''c'' is currently ''c'' = 2''e''<sup>γ</sup>.<ref>{{cite journal |first=J. |last=Pintz |title=Very large gaps between consecutive primes |journal=[[Journal of Number Theory|J. Number Theory]] |volume=63 |issue=2 |pages=286–301 |year=1997 |doi=10.1006/jnth.1997.2081 }}</ref> [[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.<ref name="Guy">Guy (2004) §A8</ref>
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| ==Conjectures about gaps between primes==
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| 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''<sub>''n''</sub>) satisfies
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| :<math> g_n = O(\sqrt{p_n} \ln p_n), </math>
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| using the [[big O notation]].
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| Later, he conjectured that the gaps are even smaller. Roughly speaking he conjectured that
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| :<math> g_n = O\left((\ln p_n)^2\right). </math>
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| At the moment, the numerical evidence seems to point in this direction. See [[Cramér's conjecture]] for more details.
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| The [[Firoozbakht’s conjecture]] which is a slight strengthening to Cramér's, satisfies
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| :<math> g_n < (\log p_{n})^2 - \log p_{n}</math>.
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| Mean while, the [[Oppermann's conjecture]] is a conjecture which is weaker than Cramér's conjecture. The expected gap size with Oppermann's conjecture is
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| :<math> g_n < \sqrt{p_n}\, </math>.
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| [[Andrica's conjecture]], which is a weaker conjecture to Oppermann's, states that<ref name="Guy"/>
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| :<math> g_n < 2\sqrt{p_n} + 1.\, </math>
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| This is a slight strengthening of [[Legendre's conjecture]] that between successive square numbers there is always a prime.
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| ==As an arithmetic function==
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| The gap ''g''<sub>''n''</sub> 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''<sub>''n''</sub> and called the prime difference function.<ref name="Guy"/> The function is neither [[multiplicative function|multiplicative]] nor [[additive function|additive]].
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| == See also ==
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| {{Portal|Mathematics}}
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| * [[Bonse's inequality]]
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| * [[Gaussian moat]]
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| * [[Twin prime]]
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| {{clear}}
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| ==References==
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| {{Reflist}}
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| * {{cite book |last=Guy | first=Richard K. | authorlink=Richard K. Guy | title=Unsolved problems in number theory | publisher=[[Springer-Verlag]] |edition=3rd | year=2004 |isbn=978-0-387-20860-2 | zbl=1058.11001 }}
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| ==External links==
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| * [[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.
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| *{{MathWorld|urlname=PrimeDifferenceFunction|title=Prime Difference Function}}
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| *{{planetmath reference|id=3143|title=Prime Difference Function}}
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| * Armin Shams, [http://link.springer.com/article/10.1007%2Fs11253-008-0034-7 Re-extending Chebyshev's theorem about Bertrand's conjecture], does not involve an 'arbitrarily big' constant as some other reported results.
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| *[[Chris Caldwell]], [http://primes.utm.edu/notes/gaps.html ''Gaps Between Primes'']; an elementary introduction
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| *[http://www.primegaps.com/ www.primegaps.com] A study of the gaps between consecutive prime numbers
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| *[[Andrew Granville]], [http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf ''Primes in Intervals of Bounded Length'']; overview of the results obtained so far up to and including James Maynard's work of November 2013.
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| {{Prime number classes}}
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| [[Category:Prime numbers]]
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| [[Category:Arithmetic functions]]
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