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From: Fell Nenwick on 19 Jul 2010 13:53
[note this is clean Non-JSH Math. ]


http://mathworld.wolfram.com/PrimeGaps.html

Prime Gaps or Cups

A prime gap of length is a run of consecutive composite numbers between
two successive primes. Therefore, the difference between two successive
primes and bounding a prime gap of length is , where is the th prime
number. Since the prime difference function
(1)

is always even (except for ), all primes gaps are also even. The notation
is commonly used to denote the smallest prime corresponding to the start of
a prime gap of length , i.e., such that is prime, , , ..., are all
composite, and is prime (with the additional constraint that no smaller
number satisfying these properties exists).
The maximal prime gap is the length of the largest prime gap that begins
with a prime less than some maximum value . For , 2, ..., is given by 4,
8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ...
(Sloane's A053303).
Arbitrarily large prime gaps exist. For example, for any , the numbers , ,
...., are all composite (Havil 2003, p. 170). However, no general method
more sophisticated than an exhaustive search is known for the determination
of first occurrences and maximal prime gaps (Nicely 1999).

Cram�r (1937) and Shanks (1964) conjectured that
(2)

Wolf conjectures a slightly different form
(3)

which agrees better with numerical evidence.
Wolf conjectures that the maximal gap between two consecutive primes less
than appears approximately at
(4)

where is the prime counting function and is the twin primes constant.
Setting reduces to Cramer's conjecture for large ,
(5)

It is known that there is a prime gap of length 803 following , and a prime
gap of length following (Baugh and O'Hara 1992). H. Dubner (2001)
discovered a prime gap of length between two 3396-digit probable primes. On
Jan. 15, 2004, J. K. Andersen and H. Rosenthal found a prime gap of length
between two probabilistic primes of digits each. In January-May 2004, Hans
Rosenthal and Jens Kruse Andersen found a prime gap of length between two
probabilistic primes with digits each (Anderson 2004).
The merit of a prime gap compares the size of a gap to the local average
gap, and is given by . In 1999, the number 1693182318746371 was found, with
merit . This remained the record merit until 804212830686677669 was found in
2005, with a gap of 1442 and a merit of . Andersen maintains a list of the
top 20 known merits. The prime gaps of increasing merit are 2, 3, 7, 113,
1129, 1327, 19609, ... (Sloane's A111870).
Young and Potler (1989) determined the first occurrences of prime gaps up to
, with all first occurrences found between 1 and 673. Nicely (1999) has
extended the list of maximal prime gaps. The following table gives the
values of for small , omitting degenerate runs which are part of a run with
greater (Sloane's A005250 and A002386).

12354
23382
47384
623394
889456
14113464
18523468
20887474
22486
34490
36500
44514
52516
72532
86534
96540
112582
114588
118602
132652
148674
154716
180766
210778
220804
222806
234906
248916
250924
282
288
292
320
336

Define
(6)

as the infimum limit of the ratio of the th prime difference to the natural
logarithm of the th prime number. If there are an infinite number of twin
primes, then . This follows since it must then be true that infinitely
often, say at for , 2, ..., so a necessary condition for the twin prime
conjecture to hold is that
(7)
(8)
(9)
(10)

However, this condition is not sufficient, since the same proof works if 2
is replaced by any constant.
Hardy and Littlewood showed in 1926 that, subject to the truth of the
generalized Riemann hypothesis, . This was subsequently improved by Rankin
(again assuming the generalized Riemann hypothesis) to . In 1940, Erdos used
sieve theory to show for the first time with no assumptions that . This was
subsequently improved to 15/16 (Ricci), (Bombieri and Davenport 1966), and
(Pil'Tai 1972), as quoted in Le Lionnais (1983, p. 26). Huxley (1973, 1977)
obtained , which was improved by Maier in 1986 to , which was the best
result known until 2003 (American Institute of Mathematics).
At a March 2003 meeting on elementary and analytic number theory in
Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof
that . While the original proof turned out to be flawed (Mackenzie 2003ab),
the result has now been established by a new proof (American Institute of
Mathematics 2005, Cipra 2005, Devlin 2005, Goldston et al. 2005ab).
SEE ALSO: Cram�r-Granville Conjecture, Jumping Champion, Nearest Prime,
Prime Constellation, Prime Difference Function, Prime Distance, Shanks'
Conjecture, Twin Primes

REFERENCES:
American Institute of Mathematics. "Small Gaps between Consecutive Primes:
Recent Work of D. Goldston and C. Yildirim."
http://www.aimath.org/goldston_tech/.
American Institute of Mathematics. "Breakthrough in Prime Number Theory."
May 24, 2005. http://aimath.org/.
Andersen, J. K. "First Known Prime Megagap."
http://hjem.get2net.dk/jka/math/primegaps/megagap.htm.
Andersen, J. K. "Largest Known Prime Gap."
http://hjem.get2net.dk/jka/math/primegaps/megagap2.htm.
Andersen, J. K. "A Prime Gap of 1001548." 15 Jan 2004.
http://listserv.nodak.edu/scripts/wa.exe?A2=ind0401&L=nmbrthry&F=&S=&P=397.
Andersen, J. K. "A Prime Gap of 2254930." 2 Jun 2004.
http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0406&L=nmbrthry&P=601.
Andersen, J. K. "Top-20 Prime Gaps."
http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm.
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.
Caldwell, C. "The Gaps Between Primes."
http://primes.utm.edu/notes/gaps.html.
Cipra, B. "Proof Promises Progress in Prime Progressions." Science 304,
1095, 2004.
Cipra, B. "Third Time Proves Charm for Prime-Gap Theorem." Science 308,
1238, 2005.
Cram�r, H. "On the Order of Magnitude of the Difference between Consecutive
Prime Numbers." Acta Arith. 2, 23-46, 1937.
Cutter, P. A. "Finding Prime Pairs with Particular Gaps." Math. Comput. 70,
1737-1744, 2001.
Devlin, K. "Major Advance on the Twin Primes Conjecture." May 24, 2005.
http://www.maa.org/news/052505twinprimes.html.
Dubner, H. "New Large Prime Gap Record." 13 Dec 2001.
http://listserv.nodak.edu/scripts/wa.exe?A2=ind0112&L=nmbrthry&P=1093.
Fouvry, �. "Autour du th�or�me de Bombieri-Vinogradov." Acta. Math. 152,
219-244, 1984.
Fouvry, �. and Grupp, F. "On the Switching Principle in Sieve Theory." J.
reine angew. Math. 370, 101-126, 1986.
Fouvry, �. and Iwaniec, H. "Primes in Arithmetic Progression." Acta Arith.
42, 197-218, 1983.
Goldston, D. A.; Graham, S. W.; Pintz, J.; and Yildirim, C. Y. "Small Gaps
between Primes or Almost Primes." Jun. 3, 2005a.
http://www.arxiv.org/abs/math.NT/0506067/.
Goldston, D. A.; Motohashi, Y.; Pintz, J.; and Yildirim, C. Y. "Small Gaps
between Primes Exist." May 14, 2005b.
http://www.arxiv.org/abs/math.NT/0505300/.
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.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton
University Press, 2003.
Huxley, M. N. "Small Differences between Consecutive Primes." Mathematica
20, 229-232, 1973.
Huxley, M. N. "Small Differences between Consecutive Primes. II."
Mathematica 24, 142-152, 1977.
Lander, L. J. and Parkin, T. R. "On First Appearance of Prime Differences."
Math. Comput. 21, 483-488, 1967.
Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.
Mackenzie, D. "Prime Proof Helps Mathematicians Mind the Gaps." Science 300,
32, 2003a.
Mackenzie, D. "Prime-Number Proof's Leap Falls Short." Science 300, 1066,
2003b.
Montgomery, H. "Small Gaps Between Primes." 13 Mar 2003.
http://listserv.nodak.edu/scripts/wa.exe?A2=ind0303&L=nmbrthry&P=1323.
Nicely, T. R. "First Occurrence Prime Gaps."
http://www.trnicely.net/gaps/gaplist.html.
Nicely, T. R. "New Maximal Prime Gaps and First Occurrences." Math. Comput.
68, 1311-1315, 1999. http://www.trnicely.net/gaps/gaps.html.
Nicely, T. R. and Nyman, B. "First Occurrence of a Prime Gap of 1000 or
Greater." http://www.trnicely.net/gaps/gaps2.html.
Nyman, B. and Nicely, T. R. "New Prime Gaps Between and ." J. Int. Seq. 6,
1-6, 2003.
Rivera, C. "Problems & Puzzles: Puzzle 011-Distinct, Increasing & Decreasing
Gaps." http://www.primepuzzles.net/puzzles/puzz_011.htm.
Shanks, D. "On Maximal Gaps between Successive Primes." Math. Comput. 18,
646-651, 1964.
Sloane, N. J. A. Sequences A002386/M0858, A008996, A008950, A008995,
A008996, A030296, A053303, and A111870 in "The On-Line Encyclopedia of
Integer Sequences."
Soundararajan, K. "Small Gaps Between Prime Numbers: The Work of
Goldston-Pintz-Yildirim." Bull. Amer. Math. Soc. 44, 1-18, 2007.
Wolf, M. "First Occurrence of a Given Gap between Consecutive Primes."
http://www.ift.uni.wroc.pl/~mwolf/.
Wolf, M. "Some Conjectures on the Gaps Between Consecutive Primes."
http://www.ift.uni.wroc.pl/~mwolf/.
Young, J. and Potler, A. "First Occurrence Prime Gaps." Math. Comput. 52,
221-224, 1989.


From: bert on 19 Jul 2010 14:53
On 19 July, 18:53, "Fell Nenwick" <inva...(a)invalid.com> wrote:
> [note this is clean Non-JSH Math. ]
>
> http://mathworld.wolfram.com/PrimeGaps.html
>
> Prime Gaps or Cups
>
> A prime gap of length  is a run of  consecutive composite numbers between
> two successive primes. Therefore, the difference between two successive
> primes  and  bounding a prime gap of length  is , where  is the th prime
> number. Since the prime difference function
> (1)
>
> is always even (except for ), all primes gaps  are also even. The notation
> is commonly used to denote the smallest prime  corresponding to the start of
> a prime gap of length , i.e., such that  is prime, , , ...,  are all
> composite, and  is prime (with the additional constraint that no smaller
> number satisfying these properties exists).
> The maximal prime gap  is the length of the largest prime gap that begins
> with a prime  less than some maximum value . For , 2, ...,  is given by 4,
> 8, 20, 36, 72, 114, 154, 220, 282, 354, 464, 540, 674, 804, 906, 1132, ....
> (Sloane's A053303).
> Arbitrarily large prime gaps exist. For example, for any , the numbers , ,
> ...,  are all composite (Havil 2003, p. 170). However, no general method
> more sophisticated than an exhaustive search is known for the determination
> of first occurrences and maximal prime gaps (Nicely 1999).
>
> Cramér (1937) and Shanks (1964) conjectured that
> (2)
>
> Wolf conjectures a slightly different form
> (3)
>
> which agrees better with numerical evidence.
> Wolf conjectures that the maximal gap  between two consecutive primes less
> than  appears approximately at
> (4)
>
> where  is the prime counting function and  is the twin primes constant.
> Setting  reduces to Cramer's conjecture for large ,
> (5)
>
> It is known that there is a prime gap of length 803 following , and a prime
> gap of length  following  (Baugh and O'Hara 1992). H. Dubner (2001)
> discovered a prime gap of length  between two 3396-digit probable primes. On
> Jan. 15, 2004, J. K. Andersen and H. Rosenthal found a prime gap of length
> between two probabilistic primes of  digits each. In January-May 2004, Hans
> Rosenthal and Jens Kruse Andersen found a prime gap of length  between two
> probabilistic primes with  digits each (Anderson 2004).
> The merit of a prime gap compares the size of a gap to the local average
> gap, and is given by . In 1999, the number 1693182318746371 was found, with
> merit . This remained the record merit until 804212830686677669 was found in
> 2005, with a gap of 1442 and a merit of . Andersen maintains a list of the
> top 20 known merits. The prime gaps of increasing merit are 2, 3, 7, 113,
> 1129, 1327, 19609, ... (Sloane's A111870).
> Young and Potler (1989) determined the first occurrences of prime gaps up to
> , with all first occurrences found between 1 and 673. Nicely (1999) has
> extended the list of maximal prime gaps. The following table gives the
> values of  for small , omitting degenerate runs which are part of a run with
> greater  (Sloane's A005250 and A002386).
>
> 12354
> 23382
> 47384
> 623394
> 889456
> 14113464
> 18523468
> 20887474
> 22486
> 34490
> 36500
> 44514
> 52516
> 72532
> 86534
> 96540
> 112582
> 114588
> 118602
> 132652
> 148674
> 154716
> 180766
> 210778
> 220804
> 222806
> 234906
> 248916
> 250924
> 282
> 288
> 292
> 320
> 336
>
> Define
> (6)
>
> as the infimum limit of the ratio of the th prime difference to the natural
> logarithm of the th prime number. If there are an infinite number of twin
> primes, then . This follows since it must then be true that  infinitely
> often, say at  for , 2, ..., so a necessary condition for the twin prime
> conjecture to hold is that
> (7)
> (8)
> (9)
> (10)
>
> However, this condition is not sufficient, since the same proof works if 2
> is replaced by any constant.
> Hardy and Littlewood showed in 1926 that, subject to the truth of the
> generalized Riemann hypothesis, . This was subsequently improved by Rankin
> (again assuming the generalized Riemann hypothesis) to . In 1940, Erdos used
> sieve theory to show for the first time with no assumptions that . This was
> subsequently improved to 15/16 (Ricci),  (Bombieri and Davenport 1966), and
> (Pil'Tai 1972), as quoted in Le Lionnais (1983, p. 26). Huxley (1973, 1977)
> obtained , which was improved by Maier in 1986 to , which was the best
> result known until 2003 (American Institute of Mathematics).
> At a March 2003 meeting on elementary and analytic number theory in
> Oberwolfach, Germany, Goldston and Yildirim presented an attempted proof
> that . While the original proof turned out to be flawed (Mackenzie 2003ab),
> the result has now been established by a new proof (American Institute of
> Mathematics 2005, Cipra 2005, Devlin 2005, Goldston et al. 2005ab).
> SEE ALSO: Cramér-Granville Conjecture, Jumping Champion, Nearest Prime,
> Prime Constellation, Prime Difference Function, Prime Distance, Shanks'
> Conjecture, Twin Primes
>
> REFERENCES:
> American Institute of Mathematics. "Small Gaps between Consecutive Primes:
> Recent Work of D. Goldston and C. Yildirim."http://www.aimath.org/goldston_tech/.
> American Institute of Mathematics. "Breakthrough in Prime Number Theory."
> May 24, 2005.http://aimath.org/.
> Andersen, J. K. "First Known Prime Megagap."http://hjem.get2net.dk/jka/math/primegaps/megagap.htm.
> Andersen, J. K. "Largest Known Prime Gap."http://hjem.get2net.dk/jka/math/primegaps/megagap2.htm.
> Andersen, J. K. "A Prime Gap of 1001548." 15 Jan 2004.http://listserv.nodak.edu/scripts/wa.exe?A2=ind0401&L=nmbrthry&F=&S=&....
> Andersen, J. K. "A Prime Gap of 2254930." 2 Jun 2004.http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0406&L=nmbrthry&P=601.
> Andersen, J. K. "Top-20 Prime Gaps."http://hjem.get2net.dk/jka/math/primegaps/gaps20.htm.
> 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.
> Caldwell, C. "The Gaps Between Primes."http://primes.utm.edu/notes/gaps.html.
> Cipra, B. "Proof Promises Progress in Prime Progressions." Science 304,
> 1095, 2004.
> Cipra, B. "Third Time Proves Charm for Prime-Gap Theorem." Science 308,
> 1238, 2005.
> Cramér, H. "On the Order of Magnitude of the Difference between Consecutive
> Prime Numbers." Acta Arith. 2, 23-46, 1937.
> Cutter, P. A. "Finding Prime Pairs with Particular Gaps." Math. Comput. 70,
> 1737-1744, 2001.
> Devlin, K. "Major Advance on the Twin Primes Conjecture." May 24, 2005.http://www.maa.org/news/052505twinprimes.html.
> Dubner, H. "New Large Prime Gap Record." 13 Dec 2001.http://listserv.nodak.edu/scripts/wa.exe?A2=ind0112&L=nmbrthry&P=1093.
> Fouvry, É. "Autour du théorème de Bombieri-Vinogradov." Acta. Math. 152,
> 219-244, 1984.
> Fouvry, É. and Grupp, F. "On the Switching Principle in Sieve Theory." J.
> reine angew. Math. 370, 101-126, 1986.
> Fouvry, É. and Iwaniec, H. "Primes in Arithmetic Progression." Acta Arith.
> 42, 197-218, 1983.
> Goldston, D. A.; Graham, S. W.; Pintz, J.; and Yildirim, C. Y. "Small Gaps
> between Primes or Almost Primes." Jun. 3, 2005a.http://www.arxiv.org/abs/math.NT/0506067/.
> Goldston, D. A.; Motohashi, Y.; Pintz, J.; and Yildirim, C. Y. "Small Gaps
> between Primes Exist." May 14, 2005b.http://www.arxiv.org/abs/math.NT/0505300/.
> 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.
> Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton
> University Press, 2003.
> Huxley, M. N. "Small Differences between Consecutive Primes." Mathematica
> 20, 229-232, 1973.
> Huxley, M. N. "Small Differences between Consecutive Primes. II."
> Mathematica 24, 142-152, 1977.
> Lander, L. J. and Parkin, T. R. "On First Appearance of Prime Differences.."
> Math. Comput. 21, 483-488, 1967.
> Le Lionnais, F. Les nombres remarquables. Paris: Hermann, 1983.
> Mackenzie, D. "Prime Proof Helps Mathematicians Mind the Gaps." Science 300,
> 32, 2003a.
> Mackenzie, D. "Prime-Number Proof's Leap Falls Short." Science 300, 1066,
> 2003b.
> Montgomery, H. "Small Gaps Between Primes." 13 Mar 2003.http://listserv.nodak.edu/scripts/wa.exe?A2=ind0303&L=nmbrthry&P=1323.
> Nicely, T. R. "First Occurrence Prime Gaps."http://www.trnicely.net/gaps/gaplist.html.
> Nicely, T. R. "New Maximal Prime Gaps and First Occurrences." Math. Comput.
> 68, 1311-1315, 1999.http://www.trnicely.net/gaps/gaps.html.
> Nicely, T. R. and Nyman, B. "First Occurrence of a Prime Gap of 1000 or
> Greater."http://www.trnicely.net/gaps/gaps2.html.
> Nyman, B. and Nicely, T. R. "New Prime Gaps Between  and ." J. Int. Seq.. 6,
> 1-6, 2003.
> Rivera, C. "Problems & Puzzles: Puzzle 011-Distinct, Increasing & Decreasing
> Gaps."http://www.primepuzzles.net/puzzles/puzz_011.htm.
> Shanks, D. "On Maximal Gaps between Successive Primes." Math. Comput. 18,
> 646-651, 1964.
> Sloane, N. J. A. Sequences A002386/M0858, A008996, A008950, A008995,
> A008996, A030296, A053303, and A111870 in "The On-Line Encyclopedia of
> Integer Sequences."
> Soundararajan, K. "Small Gaps Between Prime Numbers: The Work of
> Goldston-Pintz-Yildirim." Bull. Amer. Math. Soc. 44, 1-18, 2007.
> Wolf, M. "First Occurrence of a Given Gap between Consecutive Primes."http://www.ift.uni.wroc.pl/~mwolf/.
> Wolf, M. "Some Conjectures on the Gaps Between Consecutive Primes."http://www.ift.uni.wroc.pl/~mwolf/.
> Young, J. and Potler, A. "First Occurrence Prime Gaps." Math. Comput. 52,
> 221-224, 1989.

It was a bad idea to post that. Text from Mathworld
Wolfram doesn't display properly in a newsreader, as
I've noticed before for superscripts and subscripts.
This time it's either italic font or Greek letters.
Anyway, what you've posted is unreadable. I could
have e-mailed you privately, if you had allowed it.
--

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