Largest range between primes?

2005-08-31, 10:44

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.

R.D. Silverman · 2005-08-31, 10:52

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

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 :)

Uncwilly · 2005-08-31, 15:24

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.

mfgoode · 2005-08-31, 16:21

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

ewmayer · 2005-08-31, 18:04

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;

akruppa · 2005-08-31, 18:04

The buzzword is "prime gaps", use Google.

Alex

2006-08-18, 01:56

I know of a gap of over 750

127490705678451059912076166345994002627474713081384902130819786906217567 and
127490705678451059912076166345994002627474713081384902130819786906218321

Hope this helps

Mini-Geek · 2006-08-18, 23:51

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

akruppa · 2006-08-19, 09:06

Chebyshev said it,
We'll say it again:
There's always a prime
between n and 2n.

Alex

ewmayer · 2006-08-21, 19:54

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.

2005-08-31, 10:44	#1
Unregistered 2²·3·83 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.

Similar Threads
Thread	Thread Starter	Forum	Replies	Last Post
Largest k*2^n-1 Primes Found in 2012	Kosmaj	Riesel Prime Search	3	2013-12-27 13:25
Largest k*2^n-1 Primes in 2011	Kosmaj	Riesel Prime Search	0	2012-01-01 16:52
Largest k*2^n-1 Primes in 2010	Kosmaj	Riesel Prime Search	0	2011-01-05 04:24
Largest Simultaneous Primes	robert44444uk	Octoproth Search	4	2006-02-04 18:45
What is the largest database of all primes from 2 up?	jasong	Miscellaneous Math	6	2006-01-04 23:16

2005-08-31, 15:24	#4
Uncwilly 6809 > 6502 """"""""""""""""""" Aug 2003 101×103 Posts 24576₈ 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, 18:04	#7
akruppa "Nancy" Aug 2002 Alexandria 100110100011₂ Posts	The buzzword is "prime gaps", use Google. Alex

2006-08-18, 01:56	#8
Unregistered 1000001110₂ 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 10266₈ 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 11²×97 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.