## Prime Gaps

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.