Letting

be the Prime Difference Function, Rankin has showed that

for infinitely many and some constant (Guy 1994).

Let be the smallest Prime following or more consecutive Composite Numbers.
The largest known is

The largest known prime gap is of length 4247, occurring following (Baugh and O'Hara 1992), although this gap is almost certainly not maximal (i.e., there probably exists a smaller number having a gap of the same length following it).

Let be the smallest starting Integer for a run of consecutive Composite Numbers, also called a Composite Run. No general method other than exhaustive searching is known
for determining the first occurrence for a maximal gap, although arbitrarily large gaps exist (Nicely 1998). Cramér (1937) and
Shanks (1964) conjectured that a maximal gap of length appears at approximately
. Wolf conjectures that the
maximal gap of length appears approximately at

where is the Prime Counting Function and is the Twin Primes Constant.

The first few for , 2, ... are 4, 8, 8, 24, 24, 90, 90, 114, ... (Sloane's A030296). The following table gives the same sequence omitting degenerate runs which are part of a run with greater , and is a complete list of smallest maximal runs up to . in this table is given by Sloane's A008950, and by Sloane's A008996. The ending integers for the run corresponding to are given by Sloane's A008995. Young and Potler (1989) determined the first occurrences of prime gaps up to 72,635,119,999,997, with all first occurrences found between 1 and 673. Nicely (1998) extended the list of maximal prime gaps to a length of 915, denoting gap lengths by the difference of bounding Primes, .

1 | 4 | 319 | 2,300,942,550 |

3 | 8 | 335 | 3,842,610,774 |

5 | 24 | 353 | 4,302,407,360 |

7 | 90 | 381 | 10,726,904,660 |

13 | 114 | 383 | 20,678,048,298 |

17 | 524 | 393 | 22,367,084,960 |

19 | 888 | 455 | 25,056,082,088 |

21 | 1,130 | 463 | 42,652,618,344 |

33 | 1,328 | 467 | 127,976,334,672 |

35 | 9,552 | 473 | 182,226,896,240 |

43 | 15,684 | 485 | 241,160,024,144 |

51 | 19,610 | 489 | 297,501,075,800 |

71 | 31,398 | 499 | 303,371,455,242 |

85 | 155,922 | 513 | 304,599,508,538 |

95 | 360,654 | 515 | 416,608,695,822 |

111 | 370,262 | 531 | 461,690,510,012 |

113 | 492,114 | 533 | 614,487,453,424 |

117 | 1,349,534 | 539 | 738,832,927,928 |

131 | 1,357,202 | 581 | 1,346,294,310,750 |

147 | 2,010,734 | 587 | 1,408,695,493,610 |

153 | 4,652,354 | 601 | 1,968,188,556,461 |

179 | 17,051,708 | 651 | 2,614,941,710,599 |

209 | 20,831,324 | 673 | 7,177,162,611,713 |

219 | 47,326,694 | 715 | 13,828,048,559,701 |

221 | 122,164,748 | 765 | 19,581,334,192,423 |

233 | 189,695,660 | 777 | 42,842,283,925,352 |

247 | 191,912,784 | 803 | 90,874,329,411,493 |

249 | 387,096,134 | 805 | 171,231,342,420,521 |

281 | 436,273,010 | 905 | 218,209,405,436,543 |

287 | 1,294,268,492 | 915 | 1,189,459,969,825,483 |

291 | 1,453,168,142 |

**References**

Baugh, D. and O'Hara, F. ``Large Prime Gaps.'' *J. Recr. Math.* **24**, 186-187, 1992.

Berndt, B. C. *Ramanujan's Notebooks, Part IV.* New York: Springer-Verlag, pp. 133-134, 1994.

Bombieri, E. and Davenport, H. ``Small Differences Between Prime Numbers.'' *Proc. Roy. Soc. A* **293**, 1-18, 1966.

Brent, R. P. ``The First Occurrence of Large Gaps Between Successive Primes.'' *Math. Comput.* **27**, 959-963, 1973.

Brent, R. P. ``The Distribution of Small Gaps Between Successive Primes.'' *Math. Comput.* **28**, 315-324, 1974.

Brent, R. P. ``The First Occurrence of Certain Large Prime Gaps.'' *Math. Comput.* **35**, 1435-1436, 1980.

Cramér, H. ``On the Order of Magnitude of the Difference Between Consecutive Prime Numbers.'' *Acta Arith.* **2**, 23-46, 1937.

Guy, R. K. ``Gaps between Primes. Twin Primes'' and ``Increasing and Decreasing Gaps.'' §A8 and A11 in
*Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 19-23 and 26-27, 1994.

Lander, L. J. and Parkin, T. R. ``On First Appearance of Prime Differences.'' *Math. Comput.* **21**, 483-488, 1967.

Nicely, T. R. ``New Maximal Prime Gaps and First Occurrences.'' http://lasi.lynchburg.edu/Nicely_T/public/gaps/gaps.htm. To Appear in *Math. Comput.*

Shanks, D. ``On Maximal Gaps Between Successive Primes.'' *Math. Comput.* **18**, 646-651, 1964.

Sloane, N. J. A. Sequences A008950, A008995, A008996, and A030296 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Wolf, M. ``First Occurrence of a Given Gap Between Consecutive Primes.'' http://rose.ift.uni.wroc.pl/~mwolf.

Young, J. and Potler, A. ``First Occurrence Prime Gaps.'' *Math. Comput.* **52**, 221-224, 1989.

© 1996-9

1999-05-26