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 2005-08-31, 10:44 #1 Unregistered   2×4,561 Posts Largest range between primes? I'm curious of any large ranges between primes.. for instance, 11 and 13 have a range of 2.. which is quite small :p i'm curious of any large ranges between primes.. not necessarily the largest range, althought that would be interesting.. but just any large ranges you may know of.. I do not want to just read through the list of 10000 or more primes trying to notice some, so i'm wondering if anyone knows any off hand. Thanks.
2005-08-31, 10:52   #2
R.D. Silverman

Nov 2003

22·5·373 Posts

Quote:
 Originally Posted by Unregistered I'm curious of any large ranges between primes.. for instance, 11 and 13 have a range of 2.. which is quite small :p i'm curious of any large ranges between primes.. not necessarily the largest range, althought that would be interesting.. but just any large ranges you may know of.. I do not want to just read through the list of 10000 or more primes trying to notice some, so i'm wondering if anyone knows any off hand. Thanks.
Huh? I am not sure of the reason behind your question. The gaps
between primes can be arbitrarily large.

2005-08-31, 12:21   #3
Unregistered

9816 Posts

Quote:
 Originally Posted by R.D. Silverman Huh? I am not sure of the reason behind your question. The gaps between primes can be arbitrarily large.
no reason I was just merely curious how large the gaps can get further on down the list of primes.. i've really only looked at the first 30 or so primes.

well there is a reason kind of.. i have just recently been looking at primes and seeing if I can see any patterns.. can't really say how the range would relate to any patterns im thinking of.. im just merely curious :)

 2005-08-31, 15:24 #4 Uncwilly 6809 > 6502     """"""""""""""""""" Aug 2003 101×103 Posts 2×32×7×73 Posts I have some data on this on a different machine. In the range of smaller numbers <100,000,000,000 there are gaps of ~230 (if I recall correctly). I can post some of the actual info when I get a chance.
2005-08-31, 16:21   #5
mfgoode
Bronze Medalist

Jan 2004
Mumbai,India

22·33·19 Posts
Largest range between primes?

Quote:
 Originally Posted by Unregistered I'm curious of any large ranges between primes.. for instance, 11 and 13 have a range of 2.. which is quite small :p i'm curious of any large ranges between primes.. not necessarily the largest range, althought that would be interesting.. but just any large ranges you may know of.. I do not want to just read through the list of 10000 or more primes trying to notice some, so i'm wondering if anyone knows any off hand. Thanks.
You can have arbitrarily long sequences of consecutive numbers that are not prime.
Let n! represent the product of all whole numbers from 1 to n. Thus n! can be divided by every whole number from 2 to n.
Now construct the sequence of consecutive n! + 2, n! + 3 , n! + 4 and so on all the way to n! +n. These terms are divisible by 2 , 3 , etc. till n - 1 numbers
and so none is a prime number.
By choosing n as large as you want you can have a prime free sequence of consecutive whole numbers as long as you want
Mally

 2005-08-31, 18:04 #6 ewmayer ∂2ω=0     Sep 2002 República de California 26·181 Posts As far as actual-largest-gaps go, here is a table I lifted from an old Fortran code of mine. The (a:b) numbers following each 'case' should be read as "for integers between a and b," and the maximal gap between primes in that interval is 2*ncopy. (I believe I may have gotten the data from a table in Riesel's book): Code:  case( 2: 7); ncopy=1; case( 8: 23); ncopy=2; case( 24: 29); ncopy=3; case( 30: 97); ncopy=4; case( 98: 127); ncopy=7; case( 128: 541); ncopy=9; case( 542: 907); ncopy=10; case( 908: 1151); ncopy=11; case( 1152: 1361); ncopy=17; case( 1362: 9587); ncopy=18; case( 9588: 15727); ncopy=22; case( 15728: 19661); ncopy=26; case( 19662: 31469); ncopy=36; case( 31470: 156007); ncopy=43; case( 156008: 360749); ncopy=48; case( 360750: 370373); ncopy=56; case( 370374: 492227); ncopy=57; case( 492228: 1349651); ncopy=59; case( 1349652: 1357333); ncopy=66; case( 1357334: 2101881); ncopy=74; case( 2101882: 4652507); ncopy=77; case( 4652508: 17051887); ncopy=90; case( 17051888: 20831533); ncopy=105; case( 20831534: 47326913); ncopy=110; case( 47326914: 122164969); ncopy=111; case( 122164970: 189695893); ncopy=117; case( 189695894: 191913031); ncopy=124; case( 191913032: 387096383); ncopy=125; case( 387096384: 436273291); ncopy=141; case( 436273292: 1294268779); ncopy=144; case( 1294268780: 1453168433); ncopy=146; case( 1453168434: 2300942869); ncopy=160; case( 2300942870: 3842611109); ncopy=168; case( 3842611110: 4302407713); ncopy=177; case( 4302407714: 10726905041); ncopy=191; case( 10726905042: 20678048681); ncopy=192; case( 20678048682: 22367085353); ncopy=197; case( 22367085354: 25056082543); ncopy=228; case( 25056082544: 42652518807); ncopy=232; case( 42652518808: 127976335139); ncopy=234; case( 127976335140: 182226896713); ncopy=237; case( 182226896714: 241160624629); ncopy=243; case( 241160624630: 297501076289); ncopy=245; case( 297501076290: 303371455741); ncopy=250; case( 303371455742: 304599509051); ncopy=257; case( 304599509052: 416608696337); ncopy=258; case( 416608696338: 461690510543); ncopy=266; case( 461690510544: 614487454057); ncopy=267; case( 614487454058: 738832928467); ncopy=270; case( 738832928468: 1346294311331); ncopy=291; case( 1346294311332: 1408695494197); ncopy=294; case( 1408695494198: 1968188557063); ncopy=301; case( 1968188557064: 2614941711251); ncopy=326; case( 2614941711252: 7177162612387); ncopy=337; case( 7177162612388:13829048560417); ncopy=358; case(13829048560418:19581334193189); ncopy=383; case(19581334193190:42842283926129); ncopy=389;
 2005-08-31, 18:04 #7 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts The buzzword is "prime gaps", use Google. Alex
 2006-08-18, 01:56 #8 Unregistered   1,229 Posts I know of a gap of over 750 127490705678451059912076166345994002627474713081384902130819786906217567 and 127490705678451059912076166345994002627474713081384902130819786906218321 Hope this helps
 2006-08-18, 23:51 #9 Mini-Geek Account Deleted     "Tim Sorbera" Aug 2006 San Antonio, TX USA 17×251 Posts Although it is extrememly likely that there are many other primes between these two, if we're talking about the gap between known primes, then $2^{25,964,951}-1$ (aka M42) and $2^{30,402,457}-1$ (aka M43) (the two highest known primes) have a gap so big the difference between their number of digits is 1,335,822
 2006-08-19, 09:06 #10 akruppa     "Nancy" Aug 2002 Alexandria 2,467 Posts Chebyshev said it, We'll say it again: There's always a prime between n and 2n. Alex
 2006-08-21, 19:54 #11 ewmayer ∂2ω=0     Sep 2002 República de California 26×181 Posts I prefer to spread at least part of the credit to the original conjecturer: Bertrand proposed it, and Chebyshev proved it true, There's always a prime Between n and n times 2.

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