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Prime gap
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<h2 id="simple-observations">Simple observations</h2> <p>The first, smallest, and only <a href="/facts/Parity_(mathematics)/7zq2UM4V">odd</a> prime gap is the gap of size 1 between 2, the only <a href="/facts/Parity_(mathematics)/7zq2UM4V">even</a> prime number, and 3, the first odd prime. All other prime gaps are even. There is only one pair of consecutive gaps having length 2: the gaps <i>g</i>2 and <i>g</i>3 between the primes 3, 5, and 7. </p><p>For any <a href="/facts/Integer/PUwwrawx">integer</a> <i>n</i>, the <a href="/facts/Factorial/kkfy1Bwe">factorial</a> <i>n</i>! is the product of all positive integers up to and including <i>n</i>. Then in the sequence </p> n ! + 2 , n ! + 3 , … , n ! + n {\displaystyle n!+2,\;n!+3,\;\ldots ,\;n!+n} <p>the first term is <a href="/facts/Divisible/DKNa2GvC">divisible</a> by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of <i>n</i> − 1 consecutive <a href="/facts/Composite_number/SPtW9ftC">composite</a> integers, and it must belong to a gap between primes having length at least <i>n</i>. It follows that there are gaps between primes that are arbitrarily large, that is, for any integer <i>N</i>, there is an integer <i>m</i> with <i>g</i><i>m</i> ≥ <i>N</i>. </p><p>However, prime gaps of <i>n</i> numbers can occur at numbers much smaller than <i>n</i>!. For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000. </p><p>The average gap between primes increases as the <a href="/facts/Natural_logarithm/PnreKEzI">natural logarithm</a> of these primes, and therefore the <a href="/facts/Ratio/mFmhQt7T">ratio</a> of the prime gap to the primes involved decreases (and is asymptotically zero). This is a consequence of the <a href="/facts/Prime_number_theorem/3KA9egls">prime number theorem</a>. From a heuristic view, we expect the probability that the ratio of the length of the gap to the natural logarithm is greater than or equal to a fixed positive number <i>k</i> to be <i>e</i>−<i>k</i>; consequently the ratio can be arbitrarily large. Indeed, the ratio of the gap to the number of digits of the integers involved does increase without bound. This is a consequence of a result by Westzynthius.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> </p><p>In the opposite direction, the <a href="/facts/Twin_prime_conjecture/272VWQbc">twin prime conjecture</a> posits that <i>g</i><i>n</i> = 2 for infinitely many integers <i>n</i>. </p> <h2 id="numerical-results">Numerical results</h2> <p>Usually the ratio of g n ln ( p n ) {\textstyle {\frac {g_{n}}{\ln(p_{n})}}} is called the <i>merit</i> of the gap <i>gn</i>. Informally, the merit of a gap <i>gn</i> can be thought of as the ratio of the size of the gap compared to the average prime gap sizes in the vicinity of <i>pn</i>. </p><p>The largest known prime gap with identified <a href="/facts/Probable_prime/FTpQ3dBI">probable prime</a> gap ends has length 16,045,848, with 385,713-digit probable primes and merit <i>M</i> = 18.067, found by Andreas Höglund in March 2024.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> The largest known prime gap with identified proven primes as gap ends has length 1,113,106 and merit 25.90, with 18,662-digit primes found by P. Cami, M. Jansen and J. K. Andersen.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>As of September 2022<a href="https://en.wikipedia.org/w/index.php?title=Prime_gap&action=edit">[update]</a>, the largest known merit value and first with merit over 40, as discovered by the Gapcoin network, is 41.93878373 with the 87-digit prime 293703234068022590158723766104419463425709075574811762098588798217895728858676728143227. The prime gap between it and the next prime is 8350.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> Largest known merit values (as of October 2020<a href="https://en.wikipedia.org/w/index.php?title=Prime_gap&action=edit">[update]</a>)<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a><a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><table><tbody><tr><th>Merit</th><th><i>gn</i></th><th>digits</th><th><i>pn</i></th><th>Date</th><th>Discoverer</th></tr><tr><td>41.938784</td><td>08350</td><td>0087</td><td><i>see above</i></td><td>2017</td><td>Gapcoin</td></tr><tr><td>39.620154</td><td>15900</td><td>0175</td><td>3483347771 × 409<a href="/facts/Primorial/NqlQBLDp">#</a>/0030 − 7016</td><td>2017</td><td>Dana Jacobsen</td></tr><tr><td>38.066960</td><td>18306</td><td>0209</td><td>0650094367 × 491#/2310 − 8936</td><td>2017</td><td>Dana Jacobsen</td></tr><tr><td>38.047893</td><td>35308</td><td>0404</td><td>0100054841 × 953#/0210 − 9670</td><td>2020</td><td>Seth Troisi</td></tr><tr><td>37.824126</td><td>08382</td><td>0097</td><td>0512950801 × 229#/5610 − 4138</td><td>2018</td><td>Dana Jacobsen</td></tr></tbody></table> <p>The Cramér–Shanks–Granville ratio is the ratio of <i>gn</i> / (ln(<i>pn</i>))2.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> If we discard anomalously high values of the ratio for the primes 2, 3, 7, then the greatest known value of this ratio is 0.9206386 for the prime 1693182318746371. Other record terms can be found at <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A111943. </p><p>We say that <i>gn</i> is a <i>maximal gap</i>, if <i>gm</i> < <i>gn</i> for all <i>m</i> < <i>n</i>. As of October 2024<a href="https://en.wikipedia.org/w/index.php?title=Prime_gap&action=edit">[update]</a>, the largest known maximal prime gap has length 1676, found by Brian Kehrig. It is the 83rd maximal prime gap, and it occurs after the prime 20733746510561442863.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> Other record (maximal) gap sizes can be found in <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A005250, with the corresponding primes <i>pn</i> in <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A002386, and the values of <i>n</i> in <a href="/facts/On-Line_Encyclopedia_of_Integer_Sequences/yow6cVsU">OEIS</a>: A005669. The sequence of maximal gaps up to the <i>n</i>th prime is conjectured to have about 2 ln n {\displaystyle 2\ln n} terms.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> The 83 known maximal prime gaps<table><tbody><tr><td>Gaps 1 to 28<table><tbody><tr><th>#</th><th><i>gn</i></th><th><i>pn</i></th></tr><tr><td>1</td><td>1</td><td>2</td></tr><tr><td>2</td><td>2</td><td>3</td></tr><tr><td>3</td><td>4</td><td>7</td></tr><tr><td>4</td><td>6</td><td>23</td></tr><tr><td>5</td><td>8</td><td>89</td></tr><tr><td>6</td><td>14</td><td>113</td></tr><tr><td>7</td><td>18</td><td>523</td></tr><tr><td>8</td><td>20</td><td>887</td></tr><tr><td>9</td><td>22</td><td>1,129</td></tr><tr><td>10</td><td>34</td><td>1,327</td></tr><tr><td>11</td><td>36</td><td>9,551</td></tr><tr><td>12</td><td>44</td><td>15,683</td></tr><tr><td>13</td><td>52</td><td>19,609</td></tr><tr><td>14</td><td>72</td><td>31,397</td></tr><tr><td>15</td><td>86</td><td>155,921</td></tr><tr><td>16</td><td>96</td><td>360,653</td></tr><tr><td>17</td><td>112</td><td>370,261</td></tr><tr><td>18</td><td>114</td><td>492,113</td></tr><tr><td>19</td><td>118</td><td>1,349,533</td></tr><tr><td>20</td><td>132</td><td>1,357,201</td></tr><tr><td>21</td><td>148</td><td>2,010,733</td></tr><tr><td>22</td><td>154</td><td>4,652,353</td></tr><tr><td>23</td><td>180</td><td>17,051,707</td></tr><tr><td>24</td><td>210</td><td>20,831,323</td></tr><tr><td>25</td><td>220</td><td>47,326,693</td></tr><tr><td>26</td><td>222</td><td>122,164,747</td></tr><tr><td>27</td><td>234</td><td>189,695,659</td></tr><tr><td>28</td><td>248</td><td>191,912,783</td></tr></tbody></table></td><td>Gaps 29 to 56<table><tbody><tr><th>#</th><th><i>gn</i></th><th><i>pn</i></th></tr><tr><td>29</td><td>250</td><td>387,096,133</td></tr><tr><td>30</td><td>282</td><td>436,273,009</td></tr><tr><td>31</td><td>288</td><td>1,294,268,491</td></tr><tr><td>32</td><td>292</td><td>1,453,168,141</td></tr><tr><td>33</td><td>320</td><td>2,300,942,549</td></tr><tr><td>34</td><td>336</td><td>3,842,610,773</td></tr><tr><td>35</td><td>354</td><td>4,302,407,359</td></tr><tr><td>36</td><td>382</td><td>10,726,904,659</td></tr><tr><td>37</td><td>384</td><td>20,678,048,297</td></tr><tr><td>38</td><td>394</td><td>22,367,084,959</td></tr><tr><td>39</td><td>456</td><td>25,056,082,087</td></tr><tr><td>40</td><td>464</td><td>42,652,618,343</td></tr><tr><td>41</td><td>468</td><td>127,976,334,671</td></tr><tr><td>42</td><td>474</td><td>182,226,896,239</td></tr><tr><td>43</td><td>486</td><td>241,160,624,143</td></tr><tr><td>44</td><td>490</td><td>297,501,075,799</td></tr><tr><td>45</td><td>500</td><td>303,371,455,241</td></tr><tr><td>46</td><td>514</td><td>304,599,508,537</td></tr><tr><td>47</td><td>516</td><td>416,608,695,821</td></tr><tr><td>48</td><td>532</td><td>461,690,510,011</td></tr><tr><td>49</td><td>534</td><td>614,487,453,523</td></tr><tr><td>50</td><td>540</td><td>738,832,927,927</td></tr><tr><td>51</td><td>582</td><td>1,346,294,310,749</td></tr><tr><td>52</td><td>588</td><td>1,408,695,493,609</td></tr><tr><td>53</td><td>602</td><td>1,968,188,556,461</td></tr><tr><td>54</td><td>652</td><td>2,614,941,710,599</td></tr><tr><td>55</td><td>674</td><td>7,177,162,611,713</td></tr><tr><td>56</td><td>716</td><td>13,829,048,559,701</td></tr></tbody></table></td><td>Gaps 57 to 83<table><tbody><tr><th>#</th><th><i>gn</i></th><th><i>pn</i></th></tr><tr><td>57</td><td>766</td><td>19,581,334,192,423</td></tr><tr><td>58</td><td>778</td><td>42,842,283,925,351</td></tr><tr><td>59</td><td>804</td><td>90,874,329,411,493</td></tr><tr><td>60</td><td>806</td><td>171,231,342,420,521</td></tr><tr><td>61</td><td>906</td><td>218,209,405,436,543</td></tr><tr><td>62</td><td>916</td><td>1,189,459,969,825,483</td></tr><tr><td>63</td><td>924</td><td>1,686,994,940,955,803</td></tr><tr><td>64</td><td>1,132</td><td>1,693,182,318,746,371</td></tr><tr><td>65</td><td>1,184</td><td>43,841,547,845,541,059</td></tr><tr><td>66</td><td>1,198</td><td>55,350,776,431,903,243</td></tr><tr><td>67</td><td>1,220</td><td>80,873,624,627,234,849</td></tr><tr><td>68</td><td>1,224</td><td>203,986,478,517,455,989</td></tr><tr><td>69</td><td>1,248</td><td>218,034,721,194,214,273</td></tr><tr><td>70</td><td>1,272</td><td>305,405,826,521,087,869</td></tr><tr><td>71</td><td>1,328</td><td>352,521,223,451,364,323</td></tr><tr><td>72</td><td>1,356</td><td>401,429,925,999,153,707</td></tr><tr><td>73</td><td>1,370</td><td>418,032,645,936,712,127</td></tr><tr><td>74</td><td>1,442</td><td>804,212,830,686,677,669</td></tr><tr><td>75</td><td>1,476</td><td>1,425,172,824,437,699,411</td></tr><tr><td>76</td><td>1,488</td><td>5,733,241,593,241,196,731</td></tr><tr><td>77</td><td>1,510</td><td>6,787,988,999,657,777,797</td></tr><tr><td>78</td><td>1,526</td><td>15,570,628,755,536,096,243</td></tr><tr><td>79</td><td>1,530</td><td>17,678,654,157,568,189,057</td></tr><tr><td>80</td><td>1,550</td><td>18,361,375,334,787,046,697</td></tr><tr><td>81</td><td>1,552</td><td>18,470,057,946,260,698,231</td></tr><tr><td>82</td><td>1,572</td><td>18,571,673,432,051,830,099</td></tr><tr><td>83</td><td>1,676</td><td>20,733,746,510,561,442,863</td></tr></tbody></table></td></tr></tbody></table> <h2 id="further-results">Further results</h2> <h3>Upper bounds</h3> <p><a href="/facts/Bertrand%2527s_postulate/8z3837xk">Bertrand's postulate</a>, <a href="/facts/Mathematical_proof/7ECDrU80">proven</a> in 1852, states that there is always a prime number between <i>k</i> and 2<i>k</i>, so in particular <i>p</i><i>n</i> +1 < 2<i>pn</i>, which means <i>gn</i> < <i>pn</i> . </p><p>The <a href="/facts/Prime_number_theorem/3KA9egls">prime number theorem</a>, proven in 1896, says that the average length of the gap between a prime <i>p</i> and the next prime will asymptotically approach ln(<i>p</i>), the natural logarithm of <i>p</i>, for sufficiently large primes. The actual length of the gap might be much more or less than this. However, one can deduce from the prime number theorem that the gaps get arbitrarily smaller in proportion to the primes: the quotient </p> lim n → ∞ g n p n = 0. {\displaystyle \lim _{n\to \infty }{\frac {g_{n}}{p_{n}}}=0.} <p>In other words (by <a href="/facts/Limit_(mathematics)/j8JBhj4e">definition of a limit</a>), for every ϵ > 0 {\displaystyle \epsilon >0} , there is a number N {\displaystyle N} such that for all n > N {\displaystyle n>N} </p> g n < p n ϵ {\displaystyle g_{n}<p_{n}\epsilon } . <p><a href="/facts/Guido_Hoheisel/NzlhRShX">Hoheisel</a> (1930) was the first to show<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> a sublinear dependence; that there exists a constant <i>θ</i> < 1 such that </p> π ( x + x θ ) − π ( x ) ∼ x θ log ( x ) as x → ∞ , {\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log(x)}}{\text{ as }}x\to \infty ,} <p>hence showing that </p> g n < p n θ , {\displaystyle g_{n}<p_{n}^{\theta },} <p>for <a href="/facts/Sufficiently_large/wT7KzCyb">sufficiently large</a> <i>n</i>. </p><p>Hoheisel obtained the possible value 32999/33000 for <i>θ</i>. This was improved to 249/250 by <a href="/facts/Hans_Heilbronn/pYA3rk4T">Heilbronn</a>,<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> and to <i>θ</i> = 3/4 + <i>ε</i>, for any <i>ε</i> > 0, by <a href="/facts/Nikolai_Chudakov/yA0hNomQ">Chudakov</a>.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>A major improvement is due to <a href="/facts/Albert_Ingham/wKJXACdY">Ingham</a>,<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> who showed that for some positive constant <i>c</i>, </p> if ζ ( 1 / 2 + i t ) = O ( t c ) {\displaystyle \zeta (1/2+it)=O(t^{c})} then π ( x + x θ ) − π ( x ) ∼ x θ log ( x ) {\displaystyle \pi (x+x^{\theta })-\pi (x)\sim {\frac {x^{\theta }}{\log(x)}}} for any θ > ( 1 + 4 c ) / ( 2 + 4 c ) . {\displaystyle \theta >(1+4c)/(2+4c).} <p>Here, <i>O</i> refers to the <a href="/facts/Big_O_notation/weFFjSWg">big O notation</a>, <i>ζ</i> denotes the <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a> and <i>π</i> the <a href="/facts/Prime-counting_function/Dk1xJzsA">prime-counting function</a>. Knowing that any <i>c</i> > 1/6 is admissible, one obtains that <i>θ</i> may be any number greater than 5/8. </p><p>An immediate consequence of Ingham's result is that there is always a prime number between <i>n</i>3 and (<i>n</i> + 1)3, if <i>n</i> is sufficiently large.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> The <a href="/facts/Lindel%25C3%25B6f_hypothesis/5Hfcmrxs">Lindelöf hypothesis</a> would imply that Ingham's formula holds for <i>c</i> any positive number: but even this would not be enough to imply that there is a prime number between <i>n</i>2 and (<i>n</i> + 1)2 for <i>n</i> sufficiently large (see <a href="/facts/Legendre%2527s_conjecture/emDSyngz">Legendre's conjecture</a>). To verify this, a stronger result such as <a href="/facts/Cram%25C3%25A9r%2527s_conjecture/cGE2jTBQ">Cramér's conjecture</a> would be needed. </p><p><a href="/facts/Martin_Huxley/PlmQleFq">Huxley</a> in 1972 showed that one may choose <i>θ</i> = 7/12 = 0.583.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a> </p><p>A result, due to Baker, <a href="/facts/Glyn_Harman/Ny48C1w4">Harman</a> and <a href="/facts/J%25C3%25A1nos_Pintz/4wosTmRs">Pintz</a> in 2001, shows that <i>θ</i> may be taken to be 0.525.<a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a> </p><p>The above describes limits on <i>all</i> gaps; another are of interest is the <i>minimum</i> gap size. The <a href="/facts/Twin_prime_conjecture/272VWQbc">twin prime conjecture</a> asserts that there are always more gaps of size 2, but remains unproven. In 2005, <a href="/facts/Daniel_Goldston/ppYhNa1N">Daniel Goldston</a>, <a href="/facts/J%25C3%25A1nos_Pintz/4wosTmRs">János Pintz</a> and <a href="/facts/Cem_Y%25C4%25B1ld%25C4%25B1r%25C4%25B1m/1SxrgPGM">Cem Yıldırım</a> proved that </p> lim inf n → ∞ g n log p n = 0 {\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=0} <p>and 2 years later improved this<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> to </p> lim inf n → ∞ g n log p n ( log log p n ) 2 < ∞ . {\displaystyle \liminf _{n\to \infty }{\frac {g_{n}}{{\sqrt {\log p_{n}}}(\log \log p_{n})^{2}}}<\infty .} <p>In 2013, <a href="/facts/Yitang_Zhang/Xd2tT5uP">Yitang Zhang</a> proved that </p> lim inf n → ∞ g n < 7 ⋅ 10 7 , {\displaystyle \liminf _{n\to \infty }g_{n}<7\cdot 10^{7},} <p>meaning that there are infinitely many gaps that do not exceed 70 million.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a> A <a href="/facts/Polymath_Project/LGj7RXQ9">Polymath Project</a> collaborative effort to optimize Zhang's bound managed to lower the bound to 4680 on July 20, 2013.<a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a> In November 2013, <a href="/facts/James_Maynard_(mathematician)/3u2VgW1h">James Maynard</a> introduced a new refinement of the <a href="/facts/GPY_sieve/SRkQfHad">GPY sieve</a>, allowing him to reduce the bound to 600 and also show that the gaps between primes <i>m</i> apart are bounded for all <i>m</i>. That is, for any <i>m</i> there exists a bound <i>Δm</i> such that <i>p</i><i>n</i>+<i>m</i> − <i>pn</i> ≤ <i>Δm</i> for infinitely many <i>n</i>.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a> Using Maynard's ideas, the Polymath project improved the bound to 246;<a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a><a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> assuming the <a href="/facts/Elliott%25E2%2580%2593Halberstam_conjecture/j3nDYgm1">Elliott–Halberstam conjecture</a> and its generalized form, the bound has been reduced to 12 and 6, respectively.<a class="footnote-ref" id="fnref:27" href="#fn:27"><sup>27</sup></a> </p> <h3>Lower bounds</h3> <p>In 1931, Erik Westzynthius proved that maximal prime gaps grow more than logarithmically. That is,<a class="footnote-ref" id="fnref:28" href="#fn:28"><sup>28</sup></a> </p> lim sup n → ∞ g n log p n = ∞ . {\displaystyle \limsup _{n\to \infty }{\frac {g_{n}}{\log p_{n}}}=\infty .} <p>In 1938, <a href="/facts/Robert_Alexander_Rankin/6p4358n4">Robert Rankin</a> proved the existence of a constant <i>c</i> > 0 such that the inequality </p> g n > c log n log log n log log log log n ( log log log n ) 2 {\displaystyle g_{n}>{\frac {c\ \log n\ \log \log n\ \log \log \log \log n}{(\log \log \log n)^{2}}}} <p>holds for infinitely many values of <i>n</i>, improving the results of Westzynthius and <a href="/facts/Paul_Erd%25C5%2591s/63LoUTe1">Paul Erdős</a>. He later showed that one can take any constant <i>c</i> < <i>e</i>γ, where γ is the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>. The value of the constant <i>c</i> was improved in 1997 to any value less than 2<i>e</i>γ.<a class="footnote-ref" id="fnref:29" href="#fn:29"><sup>29</sup></a> </p><p>Paul Erdős offered a $10,000 prize for a proof or disproof that the constant <i>c</i> in the above inequality may be taken arbitrarily large.<a class="footnote-ref" id="fnref:30" href="#fn:30"><sup>30</sup></a> This was proved to be correct in 2014 by Ford–Green–Konyagin–Tao and, independently, James Maynard.<a class="footnote-ref" id="fnref:31" href="#fn:31"><sup>31</sup></a><a class="footnote-ref" id="fnref:32" href="#fn:32"><sup>32</sup></a> </p><p>The result was further improved to </p> g n > c log n log log n log log log log n log log log n {\displaystyle g_{n}>{\frac {c\ \log n\ \log \log n\ \log \log \log \log n}{\log \log \log n}}} <p>for infinitely many values of <i>n</i> by Ford–Green–Konyagin–Maynard–Tao.<a class="footnote-ref" id="fnref:33" href="#fn:33"><sup>33</sup></a> </p><p>In the spirit of Erdős' original prize, <a href="/facts/Terence_Tao/Do68JKn6">Terence Tao</a> offered US$10,000 for a proof that <i>c</i> may be taken arbitrarily large in this inequality.<a class="footnote-ref" id="fnref:34" href="#fn:34"><sup>34</sup></a> </p><p>Lower bounds for chains of primes have also been determined.<a class="footnote-ref" id="fnref:35" href="#fn:35"><sup>35</sup></a> </p> <h2 id="conjectures-about-gaps-between-primes">Conjectures about gaps between primes</h2> <p>As described above, the best proven bound on gap sizes is g n < p n 0.525 {\displaystyle g_{n}<p_{n}^{0.525}} (for n {\displaystyle n} sufficiently large; we do not worry about 5 − 3 > 3 0.525 {\displaystyle 5-3>3^{0.525}} or 29 − 23 > 23 0.525 {\displaystyle 29-23>23^{0.525}} ), but it is observed that even maximal gaps are significantly smaller than that, leading to a plethora of unproven conjectures. </p><p>The first group hypothesize that the exponent can be reduced to θ = 0.5 {\displaystyle \theta =0.5} . </p><p><a href="/facts/Legendre%2527s_conjecture/emDSyngz">Legendre's conjecture</a> that there always exists a prime between successive <a href="/facts/Square_numbers/jfakASPK">square numbers</a> implies that g n = O ( p n ) {\displaystyle g_{n}=O({\sqrt {p_{n}}})} . <a href="/facts/Andrica%2527s_conjecture/IWdlyVBv">Andrica's conjecture</a> states that<a class="footnote-ref" id="fnref:36" href="#fn:36"><sup>36</sup></a> </p> g n < 2 p n + 1. {\displaystyle g_{n}<2{\sqrt {p_{n}}}+1.} <p><a href="/facts/Oppermann%2527s_conjecture/F7mV5bq3">Oppermann's conjecture</a> makes the stronger claim that, for sufficiently large n {\displaystyle n} (probably n > 30 {\displaystyle n>30} ), </p> g n < p n . {\displaystyle g_{n}<{\sqrt {p_{n}}}.} <p>All of these remain unproved. <a href="/facts/Harald_Cram%25C3%25A9r/BWylq7t8">Harald Cramér</a> came close, proving<a class="footnote-ref" id="fnref:37" href="#fn:37"><sup>37</sup></a> that the Riemann hypothesis implies the gap <i>gn</i> satisfies </p> g n = O ( p n log p n ) , {\displaystyle g_{n}=O({\sqrt {p_{n}}}\log p_{n}),} <p>using the <a href="/facts/Big_O_notation/weFFjSWg">big O notation</a>. (In fact this result needs only the weaker <a href="/facts/Lindel%25C3%25B6f_hypothesis/5Hfcmrxs">Lindelöf hypothesis</a>, if one can tolerate an infinitesimally larger exponent.<a class="footnote-ref" id="fnref:38" href="#fn:38"><sup>38</sup></a>) </p> <p>In the same article, he conjectured that the gaps are far smaller. Roughly speaking, <a href="/facts/Cram%25C3%25A9r%2527s_conjecture/cGE2jTBQ">Cramér's conjecture</a> states that </p> g n = O ( ( log p n ) 2 ) , {\displaystyle g_{n}=O\!\left((\log p_{n})^{2}\right)\!,} <p>a <a href="/facts/Polylogarithmic/6zrJ4Ger">polylogarithmic</a> growth rate slower than any exponent θ > 0 {\displaystyle \theta >0} . </p><p>As this matches the observed growth rate of prime gaps, there are a number of similar conjectures. <a href="/facts/Firoozbakht%2527s_conjecture/A20Q88bV">Firoozbakht's conjecture</a> is slightly stronger, stating that p n 1 / n {\displaystyle p_{n}^{1/n}\!} is a <a href="/facts/Strictly_decreasing/MjQK0YL2">strictly decreasing</a> function of <i>n</i>, i.e., </p> p n + 1 1 / ( n + 1 ) < p n 1 / n for all n ≥ 1. {\displaystyle p_{n+1}^{1/(n+1)}\!<p_{n}^{1/n}{\text{ for all }}n\geq 1.} <p>If this conjecture were true, then g n < ( log p n ) 2 − log p n − 1 for all n > 9. {\displaystyle g_{n}<(\log p_{n})^{2}-\log p_{n}-1{\text{ for all }}n>9.} <a class="footnote-ref" id="fnref:39" href="#fn:39"><sup>39</sup></a><a class="footnote-ref" id="fnref:40" href="#fn:40"><sup>40</sup></a> It implies a strong form of Cramér's conjecture but is inconsistent with the heuristics of <a href="/facts/Andrew_Granville/EhFz6kyI">Granville</a> and <a href="/facts/J%25C3%25A1nos_Pintz/4wosTmRs">Pintz</a><a class="footnote-ref" id="fnref:41" href="#fn:41"><sup>41</sup></a><a class="footnote-ref" id="fnref:42" href="#fn:42"><sup>42</sup></a><a class="footnote-ref" id="fnref:43" href="#fn:43"><sup>43</sup></a> which suggest that g n > 2 − ε e γ ( log p n ) 2 > ( 1.1229 − ε ) ( log p n ) 2 {\displaystyle g_{n}>{\frac {2-\varepsilon }{e^{\gamma }}}(\log p_{n})^{2}>(1.1229-\varepsilon )(\log p_{n})^{2}} infinitely often for any ε > 0 , {\displaystyle \varepsilon >0,} where γ {\displaystyle \gamma } denotes the <a href="/facts/Euler%25E2%2580%2593Mascheroni_constant/2kEgRvjC">Euler–Mascheroni constant</a>. </p><p><a href="/facts/Polignac%2527s_conjecture/x0ndzGkg">Polignac's conjecture</a> states that every positive even number <i>k</i> occurs as a prime gap infinitely often. The case <i>k</i> = 2 is the <a href="/facts/Twin_prime_conjecture/272VWQbc">twin prime conjecture</a>. The conjecture has not yet been proven or disproven for any specific value of <i>k</i>, but the improvements on Zhang's result discussed above prove that it is true for at least one (currently unknown) value of <i>k</i> ≤ 246. </p> <h2 id="as-an-arithmetic-function">As an arithmetic function</h2> <p>The gap <i>g</i><i>n</i> between the <i>n</i>th and (<i>n</i> + 1)st prime numbers is an example of an <a href="/facts/Arithmetic_function/hULxwsgm">arithmetic function</a>. In this context it is usually denoted <i>d</i><i>n</i> and called the prime difference function.<a class="footnote-ref" id="fnref:44" href="#fn:44"><sup>44</sup></a> The function is neither <a href="/facts/Multiplicative_function/4kfolbBl">multiplicative</a> nor <a href="/facts/Additive_function/J5XLJSz4">additive</a>. </p> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/Bonse%2527s_inequality/2zoY9lIQ">Bonse's inequality</a></li> <li><a href="/facts/Gaussian_moat/QqYGTRIB">Gaussian moat</a></li> <li><a href="/facts/Twin_prime/272VWQbc">Twin prime</a></li></ul> <ul><li><a href="/facts/Richard_K._Guy/shtfXH4N">Guy, Richard K.</a> (2004). <i>Unsolved problems in number theory</i> (3rd ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-20860-2. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1058.11001">1058.11001</a>.</li></ul> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Kannan_Soundararajan/NVX5hAdm">Soundararajan, Kannan</a> (2007). "Small gaps between prime numbers: the work of Goldston-Pintz-Yıldırım". <i>Bull. Am. Math. Soc</i>. New Series. 44 (1): 1–18. <a href="/facts/ArXiv_(identifier)/H6EtgnBe">arXiv</a>:<a href="https://arxiv.org/abs/math/0605696">math/0605696</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1090%2Fs0273-0979-06-01142-6">10.1090/s0273-0979-06-01142-6</a>. <a href="/facts/S2CID_(identifier)/ldJsHa2Y">S2CID</a> <a href="https://api.semanticscholar.org/CorpusID:119611838">119611838</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1193.11086">1193.11086</a>.</li> <li><a href="/facts/Preda_Mih%25C4%2583ilescu/yIuayGhd">Mihăilescu, Preda</a> (June 2014). <a href="http://www.ems-ph.org/journals/newsletter/pdf/2014-06-92.pdf">"On some conjectures in additive number theory"</a> (PDF). <i>EMS Newsletter</i> (92): 13–16. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4171%2FNEWS">10.4171/NEWS</a>. <a href="/facts/Hdl_(identifier)/rdebSxmC">hdl</a>:<a href="https://hdl.handle.net/2117%2F17085">2117/17085</a>. <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1027-488X">1027-488X</a>.</li></ul> <h2 id="external-links">External links</h2> <ul><li>Thomas R. Nicely, <a href="https://faculty.lynchburg.edu/~nicely/">Some Results of Computational Research in Prime Numbers -- Computational Number Theory</a>. This reference web site includes a list of all first known occurrence prime gaps.</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/PrimeDifferenceFunction.html">"Prime Difference Function"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li> <li><a href="https://planetmath.org/PrimeDifferenceFunction">"Prime Difference Function"</a>. <i><a href="/facts/PlanetMath/wphPCtLf">PlanetMath</a></i>.</li> <li>Armin Shams, <a href="https://link.springer.com/article/10.1007%2Fs11253-008-0034-7">Re-extending Chebyshev's theorem about Bertrand's conjecture</a>, does not involve an 'arbitrarily big' constant as some other reported results.</li> <li><a href="/facts/Chris_Caldwell_(mathematician)/d1pM4IgB">Chris Caldwell</a>, <a href="http://primes.utm.edu/notes/gaps.html"><i>Gaps Between Primes</i></a>; an elementary introduction</li> <li><a href="/facts/Andrew_Granville/EhFz6kyI">Andrew Granville</a>, <a href="http://www.dms.umontreal.ca/~andrew/CEBBrochureFinal.pdf"><i>Primes in Intervals of Bounded Length</i></a>; overview of the results obtained so far up to and including James Maynard's work of November 2013.</li> <li>Birke Heeren, <a href="https://prime-numbers.de">[1]</a> Here you find the large prime number gaps and a paper on how to calculate such large gaps.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helsingsfors (in German), 5: 1–37, JFM 57.0186.02, Zbl 0003.24601. <a href="/wiki/JFM_(identifier)" target="_blank">/wiki/JFM_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>ATH (March 11, 2024). "Announcement at Mersenneforum.org". Mersenneforum.org. Archived from the original on March 12, 2024. <a href="https://mersenneforum.org/showpost.php?p=652565&postcount=300" target="_blank">https://mersenneforum.org/showpost.php?p=652565&postcount=300</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Andersen, Jens Kruse. "The Top-20 Prime Gaps". Archived from the original on December 27, 2019. Retrieved June 13, 2014. <a href="http://primerecords.dk/primegaps/gaps20.htm" target="_blank">http://primerecords.dk/primegaps/gaps20.htm</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Andersen, Jens Kruse (March 8, 2013). "A megagap with merit 25.9". primerecords.dk. Archived from the original on December 25, 2019. Retrieved September 29, 2022. <a href="http://primerecords.dk/primegaps/gap1113106.htm" target="_blank">http://primerecords.dk/primegaps/gap1113106.htm</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Nicely, Thomas R. (2019). "NEW PRIME GAP OF MAXIMUM KNOWN MERIT". faculty.lynchburg.edu. Archived from the original on April 30, 2021. Retrieved September 29, 2022. <a href="https://faculty.lynchburg.edu/~nicely/#MaxMerit" target="_blank">https://faculty.lynchburg.edu/~nicely/#MaxMerit</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>"Prime Gap Records". GitHub. June 11, 2022. <a href="https://github.com/primegap-list-project/prime-gap-list" target="_blank">https://github.com/primegap-list-project/prime-gap-list</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Nicely, Thomas R. (2019). "NEW PRIME GAP OF MAXIMUM KNOWN MERIT". faculty.lynchburg.edu. Archived from the original on April 30, 2021. Retrieved September 29, 2022. <a href="https://faculty.lynchburg.edu/~nicely/#MaxMerit" target="_blank">https://faculty.lynchburg.edu/~nicely/#MaxMerit</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>"Record prime gap info". ntheory.org. Archived from the original on October 13, 2016. Retrieved September 29, 2022. <a href="http://ntheory.org/gaps/stats.pl" target="_blank">http://ntheory.org/gaps/stats.pl</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Nicely, Thomas R. (2019). "TABLES OF PRIME GAPS". faculty.lynchburg.edu. Archived from the original on November 27, 2020. Retrieved September 29, 2022. <a href="https://faculty.lynchburg.edu/~nicely/index.html#TPG" target="_blank">https://faculty.lynchburg.edu/~nicely/index.html#TPG</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>"Top 20 overall merits". Prime gap list. Archived from the original on July 27, 2022. Retrieved September 29, 2022. <a href="https://primegap-list-project.github.io/lists/top20-overall-merits/" target="_blank">https://primegap-list-project.github.io/lists/top20-overall-merits/</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Nicely, Thomas R. (2019). "NEW PRIME GAP OF MAXIMUM KNOWN MERIT". faculty.lynchburg.edu. Archived from the original on April 30, 2021. Retrieved September 29, 2022. <a href="https://faculty.lynchburg.edu/~nicely/#MaxMerit" target="_blank">https://faculty.lynchburg.edu/~nicely/#MaxMerit</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Andersen, Jens Kruse. "Record prime gaps". Retrieved October 10, 2024. <a href="https://www.pzktupel.de/JensKruseAndersen/risinggap.php" target="_blank">https://www.pzktupel.de/JensKruseAndersen/risinggap.php</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Kourbatov, A.; Wolf, M. (2020). "On the first occurrences of gaps between primes in a residue class". Journal of Integer Sequences. 23 (Article 20.9.3). arXiv:2002.02115. MR 4167933. S2CID 211043720. Zbl 1444.11191. Archived from the original on April 12, 2021. Retrieved December 3, 2020. <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Wolf/wolf2.html" target="_blank">https://cs.uwaterloo.ca/journals/JIS/VOL23/Wolf/wolf2.html</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Hoheisel, G. (1930). "Primzahlprobleme in der Analysis". Sitzunsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin. 33: 3–11. JFM 56.0172.02. <a href="/wiki/JFM_(identifier)" target="_blank">/wiki/JFM_(identifier)</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Heilbronn, H. A. (1933). "Über den Primzahlsatz von Herrn Hoheisel". Mathematische Zeitschrift. 36 (1): 394–423. doi:10.1007/BF01188631. JFM 59.0947.01. S2CID 123216472. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Tchudakoff, N. G. (1936). "On the difference between two neighboring prime numbers". Mat. Sb. 1: 799–814. Zbl 0016.15502. <a href="/wiki/Zbl_(identifier)" target="_blank">/wiki/Zbl_(identifier)</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>Ingham, A. E. (1937). "On the difference between consecutive primes". Quarterly Journal of Mathematics. Oxford Series. 8 (1): 255–266. Bibcode:1937QJMat...8..255I. doi:10.1093/qmath/os-8.1.255. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Cheng, Yuan-You Fu-Rui (2010). "Explicit estimate on primes between consecutive cubes". Rocky Mt. J. Math. 40: 117–153. arXiv:0810.2113. doi:10.1216/rmj-2010-40-1-117. S2CID 15502941. Zbl 1201.11111. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Huxley, M. N. (1972). "On the Difference between Consecutive Primes". Inventiones Mathematicae. 15 (2): 164–170. Bibcode:1971InMat..15..164H. doi:10.1007/BF01418933. S2CID 121217000. <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Baker, R. C.; Harman, G.; Pintz, J. (2001). "The difference between consecutive primes, II" (PDF). Proceedings of the London Mathematical Society. 83 (3): 532–562. CiteSeerX 10.1.1.360.3671. doi:10.1112/plms/83.3.532. S2CID 8964027. <a href="https://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf" target="_blank">https://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>Goldston, Daniel A.; Pintz, János; Yıldırım, Cem Yalçin (2010). "Primes in Tuples II". Acta Mathematica. 204 (1): 1–47. arXiv:0710.2728. doi:10.1007/s11511-010-0044-9. S2CID 7993099. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Zhang, Yitang (2014). "Bounded gaps between primes". Annals of Mathematics. 179 (3): 1121–1174. doi:10.4007/annals.2014.179.3.7. MR 3171761. <a href="/wiki/Yitang_Zhang" target="_blank">/wiki/Yitang_Zhang</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>"Bounded gaps between primes". Polymath. Archived from the original on February 28, 2020. Retrieved July 21, 2013. <a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes" target="_blank">http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Maynard, James (January 2015). "Small gaps between primes". Annals of Mathematics. 181 (1): 383–413. arXiv:1311.4600. doi:10.4007/annals.2015.181.1.7. MR 3272929. S2CID 55175056. <a href="/wiki/James_Maynard_(mathematician)" target="_blank">/wiki/James_Maynard_(mathematician)</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>"Bounded gaps between primes". Polymath. Archived from the original on February 28, 2020. Retrieved July 21, 2013. <a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes" target="_blank">http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>D.H.J. Polymath (2014). "Variants of the Selberg sieve, and bounded intervals containing many primes". Research in the Mathematical Sciences. 1 (12). arXiv:1407.4897. doi:10.1186/s40687-014-0012-7. MR 3373710. S2CID 119699189. <a href="https://doi.org/10.1186%2Fs40687-014-0012-7" target="_blank">https://doi.org/10.1186%2Fs40687-014-0012-7</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> <li id="fn:27"><p>"Bounded gaps between primes". Polymath. Archived from the original on February 28, 2020. Retrieved July 21, 2013. <a href="http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes" target="_blank">http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></p></li> <li id="fn:28"><p>Westzynthius, E. (1931), "Über die Verteilung der Zahlen die zu den n ersten Primzahlen teilerfremd sind", Commentationes Physico-Mathematicae Helsingsfors (in German), 5: 1–37, JFM 57.0186.02, Zbl 0003.24601. <a href="/wiki/JFM_(identifier)" target="_blank">/wiki/JFM_(identifier)</a> <a href="#fnref:28" class="footnote-back-ref">↩</a></p></li> <li id="fn:29"><p>Pintz, J. (1997). "Very large gaps between consecutive primes". J. Number Theory. 63 (2): 286–301. doi:10.1006/jnth.1997.2081. <a href="https://doi.org/10.1006%2Fjnth.1997.2081" target="_blank">https://doi.org/10.1006%2Fjnth.1997.2081</a> <a href="#fnref:29" class="footnote-back-ref">↩</a></p></li> <li id="fn:30"><p>Erdős, Paul; Bollobás, Béla; Thomason, Andrew, eds. (1997). Combinatorics, Geometry and Probability: A Tribute to Paul Erdős. Cambridge University Press. p. 1. ISBN 9780521584722. Archived from the original on September 29, 2022. Retrieved September 29, 2022. <a href="9780521584722" target="_blank">9780521584722</a> <a href="#fnref:30" class="footnote-back-ref">↩</a></p></li> <li id="fn:31"><p>Ford, Kevin; Green, Ben; Konyagin, Sergei; Tao, Terence (2016). "Large gaps between consecutive prime numbers". Ann. of Math. 183 (3): 935–974. arXiv:1408.4505. doi:10.4007/annals.2016.183.3.4. MR 3488740. S2CID 16336889. <a href="/wiki/Ann._of_Math." target="_blank">/wiki/Ann._of_Math.</a> <a href="#fnref:31" class="footnote-back-ref">↩</a></p></li> <li id="fn:32"><p>Maynard, James (2016). "Large gaps between primes". Ann. of Math. 183 (3): 915–933. arXiv:1408.5110. doi:10.4007/annals.2016.183.3.3. MR 3488739. S2CID 119247836. <a href="/wiki/Ann._of_Math." target="_blank">/wiki/Ann._of_Math.</a> <a href="#fnref:32" class="footnote-back-ref">↩</a></p></li> <li id="fn:33"><p>Ford, Kevin; Green, Ben; Konyagin, Sergei; Maynard, James; Tao, Terence (2018). "Long gaps between primes". J. Amer. Math. Soc. 31 (1): 65–105. arXiv:1412.5029. doi:10.1090/jams/876. MR 3718451. S2CID 14487001. <a href="/wiki/J._Amer._Math._Soc." target="_blank">/wiki/J._Amer._Math._Soc.</a> <a href="#fnref:33" class="footnote-back-ref">↩</a></p></li> <li id="fn:34"><p>Tao, Terence (December 16, 2014). "Long gaps between primes / What's new". Archived from the original on June 9, 2019. Retrieved August 29, 2019. <a href="https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/" target="_blank">https://terrytao.wordpress.com/2014/12/16/long-gaps-between-primes/</a> <a href="#fnref:34" class="footnote-back-ref">↩</a></p></li> <li id="fn:35"><p>Ford, Kevin; Maynard, James; Tao, Terence (October 13, 2015). "Chains of large gaps between primes". arXiv:1511.04468 [math.NT]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:35" class="footnote-back-ref">↩</a></p></li> <li id="fn:36"><p>Guy (2004) §A8 <a href="#fnref:36" class="footnote-back-ref">↩</a></p></li> <li id="fn:37"><p>Cramér, Harald (1936). "On the order of magnitude of the difference between consecutive prime numbers". Acta Arithmetica. 2: 23–46. doi:10.4064/aa-2-1-23-46. <a href="https://doi.org/10.4064%2Faa-2-1-23-46" target="_blank">https://doi.org/10.4064%2Faa-2-1-23-46</a> <a href="#fnref:37" class="footnote-back-ref">↩</a></p></li> <li id="fn:38"><p>Ingham, Albert E. (1937). "On the difference between consecutive primes" (PDF). Quarterly Journal of Mathematics. 8 (1). Oxford: 255–266. Bibcode:1937QJMat...8..255I. doi:10.1093/qmath/os-8.1.255. Archived (PDF) from the original on December 5, 2022. <a href="/wiki/Albert_Ingham" target="_blank">/wiki/Albert_Ingham</a> <a href="#fnref:38" class="footnote-back-ref">↩</a></p></li> <li id="fn:39"><p>Sinha, Nilotpal Kanti (2010). "On a new property of primes that leads to a generalization of Cramer's conjecture". arXiv:1010.1399 [math.NT]. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:39" class="footnote-back-ref">↩</a></p></li> <li id="fn:40"><p>Kourbatov, Alexei (2015). "Upper bounds for prime gaps related to Firoozbakht's conjecture". Journal of Integer Sequences. 18 (11) 15.11.2. arXiv:1506.03042. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:40" class="footnote-back-ref">↩</a></p></li> <li id="fn:41"><p>Granville, Andrew (1995). "Harald Cramér and the distribution of prime numbers" (PDF). Scandinavian Actuarial Journal. 1: 12–28. CiteSeerX 10.1.1.129.6847. doi:10.1080/03461238.1995.10413946. Archived (PDF) from the original on September 23, 2015. Retrieved March 2, 2016.. <a href="/wiki/Andrew_Granville" target="_blank">/wiki/Andrew_Granville</a> <a href="#fnref:41" class="footnote-back-ref">↩</a></p></li> <li id="fn:42"><p>Granville, Andrew (1995). "Unexpected Irregularities in the Distribution of Prime Numbers" (PDF). Proceedings of the International Congress of Mathematicians. Vol. 1. pp. 388–399. doi:10.1007/978-3-0348-9078-6_32. ISBN 978-3-0348-9897-3. Archived (PDF) from the original on May 7, 2016. Retrieved March 2, 2016.. <a href="978-3-0348-9897-3" target="_blank">978-3-0348-9897-3</a> <a href="#fnref:42" class="footnote-back-ref">↩</a></p></li> <li id="fn:43"><p>Pintz, János (September 2007). "Cramér vs. Cramér: On Cramér's probabilistic model for primes". Functiones et Approximatio Commentarii Mathematici. 37 (2): 232–471. doi:10.7169/facm/1229619660. <a href="/wiki/J%C3%A1nos_Pintz" target="_blank">/wiki/J%C3%A1nos_Pintz</a> <a href="#fnref:43" class="footnote-back-ref">↩</a></p></li> <li id="fn:44"><p>Guy (2004) §A8 <a href="#fnref:44" class="footnote-back-ref">↩</a></p></li> </ol>