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Old 2005-08-31, 10:44   #1
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Default 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.
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Old 2005-08-31, 10:52   #2
R.D. Silverman
 
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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.
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Old 2005-08-31, 12:21   #3
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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 :)
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Old 2005-08-31, 15:24   #4
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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.
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Old 2005-08-31, 16:21   #5
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Exclamation 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
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Old 2005-08-31, 18:04   #6
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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;
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Old 2005-08-31, 18:04   #7
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The buzzword is "prime gaps", use Google.

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Old 2006-08-18, 01:56   #8
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I know of a gap of over 750

127490705678451059912076166345994002627474713081384902130819786906217567 and
127490705678451059912076166345994002627474713081384902130819786906218321

Hope this helps
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Old 2006-08-18, 23:51   #9
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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
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Old 2006-08-19, 09:06   #10
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Chebyshev said it,
We'll say it again:
There's always a prime
between n and 2n.

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Old 2006-08-21, 19:54   #11
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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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