20050831, 10:44  #1 
7F4_{16} 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. 
20050831, 10:52  #2  
"Bob Silverman"
Nov 2003
North of Boston
2^{3}·3·311 Posts 
Quote:
between primes can be arbitrarily large. 

20050831, 12:21  #3  
3×7×431 Posts 
Quote:
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 :) 

20050831, 15:24  #4 
6809 > 6502
"""""""""""""""""""
Aug 2003
101×103 Posts
2994_{16} 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. 
20050831, 16:21  #5  
Bronze Medalist
Jan 2004
Mumbai,India
100000000100_{2} Posts 
Largest range between primes?
Quote:
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 

20050831, 18:04  #6 
∂^{2}ω=0
Sep 2002
República de California
3×7×13×43 Posts 
As far as actuallargestgaps 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; 
20050831, 18:04  #7 
"Nancy"
Aug 2002
Alexandria
2,467 Posts 
The buzzword is "prime gaps", use Google.
Alex 
20060818, 01:56  #8 
2^{2}·37 Posts 
I know of a gap of over 750
127490705678451059912076166345994002627474713081384902130819786906217567 and 127490705678451059912076166345994002627474713081384902130819786906218321 Hope this helps 
20060818, 23:51  #9 
Account Deleted
"Tim Sorbera"
Aug 2006
San Antonio, TX USA
2·3·23·31 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 (aka M42) and (aka M43) (the two highest known primes) have a gap so big the difference between their number of digits is 1,335,822

20060819, 09:06  #10 
"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 
20060821, 19:54  #11 
∂^{2}ω=0
Sep 2002
República de California
3·7·13·43 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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