## Re: [PrimeNumbers] Is there another prime gap...

Expand Messages
• ... The unusuality could be measured dividing the gap length by log(p)*log(p)*sqrt(log(log(p))), where p is the prime after the gap. Thus, the most unusual
Message 1 of 6 , Feb 5, 2008
> The fact that the gap between 1327 and 1361 is the most persistent primal gap of all, not
> being equalled until between 8467 and 8501 makes me think of it as the most unusual
> prime gap that probably exists among numbers of any size.

The "unusuality" could be measured dividing the gap length by log(p)*log(p)*sqrt(log(log(p))), where p is the prime after the gap. Thus, the most "unusual" gaps (with unusuality larger than 0.45) are:

6 between 23 and 29
14 between 113 and 127
34 between 1327 and 1361
1132 between 1693182318746371 and 1693182318747503

I don't count the gap of 4 between 7 and 11, the gap of 2 between 3 and 5 and the gap of 1 between 2 and 3, because log(log(p)) becomes inadmissibly small here.

Note that the gap of 1132 is more unusual than the gap of 34, and, who knows, it may appear the most unusual gap at all!

P.S. The factorization of the product of all 1131 composites in this gap looks as
2^1127*3^568*5^285*7^189*11^112*13^94*17^70*19^63*23^51*29^40*31^37*37^31*41^27*43^27*47^24*53^23*59^20*61^19*67^18*71^16*73^15*79^15*83^15*89^14*97^12*101^11*103^11*107^11*109^10*113^10*127^9*131^9*137^9*139^8*149^7*151^9*157^7*163^6*167^7*173^6*179^7*181^6*191^5*193^6*197^6*199^5*211^5*223^5*227^5*229^5*233^5*239^4*241^4*251^4*257^5*263^5*269^4*271^5*277^5*281^4*283^4*293^4*307^3*311^3*313^4*317^3*331^4*337^4*347^3*349^3*353^3*359^3*367^3*373^3*379^3*383^3*389^3*397^3*401^3*409^3*419^3*421^2*431^2*433^4*439^2*443^2*449^2*457^2*461^3*463^2*467^3*479^2*487^3*491^3*499^2*503^2*509^2*521^2*523^2*541^2*547^2*557^2*563^2*569^2*571^2*577^2*587^2*593^2*599^2*601^2*607^2*613^2*617^2*619^2*631*641^2*643^2*647^2*653^2*659^2*661*673^2*677*683^2*691^2*701*709^2*719*727^2*733^2*739^2*743^2*751*757^2*761^2*769*773^2*787^2*797^2*809*811^2*821^2*823^2*827^2*829^2*839*853*857^2*859^2*863*877^2*881*883*887^2*907*911*919*929*937*941*947*953*967*971*977*983*991*997^2*1009*1013*1019*1021*1031*1033*1039*1049*1051*1061*1063*1069*1087*1091*1093*1097*1103*1109*1117*1123*1129*1151*1153*1163*1171*1181*1187*1193*1201*1217*1223*1229*1231*1237*1249*1259*1277*1279*1283*1289*1291*1297*1301*1303*1307*1319*1321*1327*1367*1373*1381*1399*1423*1427*1433*1439*1447*1451*1453*1471*1481*1483*1487*1489*1493*1499*1511*1543*1549*1553*1567*1571*1583*1597*1601*1607*1609*1613*1619*1627*1637*1657*1667*1669*1693*1699*1709*1733*1741*1753*1783*1787*1801*1811*1823*1861*1867*1871*1877*1879*1889*1901*1907*1931*1933*1949*1973*1979*1987*1993*1997*1999*2017*2027*2039*2069*2083*2111*2129*2131*2143*2153*2161*2213*2239*2243*2251*2269*2273*2281*2287*2311*2333*2339*2341*2351*2357*2371*2383*2389*2411*2417*2423*2447*2503*2521*2543*2551*2591*2593*2617*2621*2633*2647*2663*2671*2683*2693*2729*2741*2797*2801*2833*2851*2917*2971*2999*3011*3041*3049*3067*3119*3181*3187*3191*3221*3253*3299*3319*3331*3343*3371*3391*3407*3433*3449*3457*3461*3467*3499*3533*3583*3607*3613*3617*3659*3671*3697*3719*3733*3739*3769*3833*3847*3851*3889*3911*3919*4051*4091*4099*4127*4139*4177*4253*4271*4327*4391*4421*4423*4441*4447*4457*4483*4567*4591*4721*4729*4787*4793*4799*4877*4937*4967*4987*5021*5101*5153*5167*5189*5351*5417*5441*5479*5483*5501*5563*5569*5573*5623*5647*5717*5737*5743*5801*5861*5881*5939*5981*6089*6091*6199*6247*6277*6547*6571*6637*6679*6703*6709*6793*6841*6871*6899*6907*6949*6983*7043*7109*7129*7151*7177*7229*7321*7369*7457*7517*7577*7603*7757*8087*8117*8171*8387*8431*8467*8543*8669*8741*8837*8933*8969*8999*9103*9323*9413*9463*9521*9613*9619*9697*9733*9829*9851*10139*10177*10181*10337*10357*10501*10513*10529*10531*10567*10601*10657*10709*10781*10889*10949*10973*10987*10993*11087*11113*11177*11273*11443*11813*11831*11867*11959*11971*12269*12289*12409*12517*12539*12757*12823*12973*13009*13457*13499*13523*13709*13721*14029*14153*14461*14717*14779*14783*14813*14891*15139*15349*15401*15511*15649*15787*15809*16217*16267*16301*16607*16649*16661*16747*16943*16963*17027*17029*17209*17393*17477*17807*17929*17977*17987*18119*18257*18311*18379*19013*19139*19207*19531*20359*20719*20731*20749*20771*20807*20921*21139*21157*21277*21649*21673*21787*22031*22073*22193*22229*22571*22691*22769*22901*22961*23053*23291*23581*23599*23869*23957*23981*24061*24229*24691*24749*24763*24793*25439*25463*25633*25639*25703*26267*26699*26921*27011*27127*27281*27361*27527*27541*28607*28879*29201*29387*29429*29527*29671*30941*31039*31159*31391*31393*31481*31567*31573*31957*32359*32363*32843*33181*33329*33647*33829*34171*34429*34607*34667*34703*34841*35053*35227*35449*35597*36241*36263*37273*37423*37537*37657*37691*38069*38699*38917*39499*39901*40039*40127*40163*40993*41333*41941*41947*42019*42187*42443*42989*43633*43997*44777*44893*45949*46187*46273*47189*47563*47741*47791*48049*49603*50069*50849*51479*51787*52021*52121*52391*53239*53401*54139*54293*54583*55259*56543*56747*57107*57367*57373*57601*58789*59447*59699*61297*61657*62969*63029*63737*63803*63977*64399*64601*66047*67021*67073*67139*68633*69767*71129*71429*71537*73277*73571*76667*76943*77167*78059*78607*78643*79103*79579*80273*80557*82031*83207*84053*84631*87281*87509*88589*89003*91807*92899*93307*93581*95813*96797*97301*98947*99907*100559*101323*102217*102481*105251*105517*106163*110947*112799*113453*114761*115421*116491*118369*121013*121349*127681*128509*129263*129629*130003*130651*131101*132437*132851*133709*134081*136403*136987*138727*139709*139987*141601*141629*146221*146347*148429*149249*151817*151841*152111*154073*154409*154681*158261*163393*165559*165817*168353*172981*174257*175709*176159*176419*177623*177841*181607*181813*183809*185483*188189*194101*198337*199211*200689*203341*207029*211339*215249*219937*222461*227191*228251*230611*233069*233419*238759*238943*242591*246707*246811*247697*251897*252877*254377*263647*265957*267593*269237*271501*276517*277021*277789*280037*283873*284819*292793*293473*295247*296969*299191*299903*305867*308809*308851*311153*311447*316499*317857*319567*323413*323789*324953*329419*332567*336103*348709*353201*354791*361469*362221*362419*364643*367949*371087*373463*374083*380929*386437*387977*388231*389629*389723*392599*404429*404693*405577*408137*425101*435649*442789*452531*454039*454507*455269*474709*476023*478901*481681*486181*493657*495589*501503*504767*507077*509801*511831*514001*518239*521021*526739*530063*531989*558539*559067*564491*566701*572687*580711*581947*596419*604729*632669*635519*640049*647963*687031*696523*698039*699191*701479*702817*705161*709901*727471*746797*749773*756853*761347*763027*764893*765287*779231*780223*780347*794557*796921*808523*811501*813811*814199*823457*833737*843607*844867*844891*850567*869863*880283*909901*912047*931981*939613*945341*951151*955613*967229*974657*976951*979177*984913*...

[Non-text portions of this message have been removed]
• I have had a look at your factorisation of all the 1132 composites after 1693182318746371 and it still does not seem as efficient in using all the primes
Message 2 of 6 , Feb 6, 2008
I have had a look at your factorisation of all the 1132 composites after
1693182318746371 and it still does not seem as "efficient" in using all the primes below
the square root of that sixteen-digit number. Have you thought of, as I did in my
demonstration, of excluding all the numbers it that gap that are divisible by 2, 3 or 5 and
then comparing the factors?

As it was I found 975 primes less than 975,000 occurring as factors, and that is as little as
1.27 percent of all the primes over the duration. This is a far cry from a situation where
every possible prime less than the square root of a number occurs! Even with the relativel
persistent gap from 31397 to 31469 14 primes of 36 below the square root of 31469 are
required to divide all those numbers not divisible by 2, 3 or 5.

I am really curious as to how much less probable it becomes that every potential prime
factor could occur over an unusually large prime gap. I do imagine that there does exist a
difficulty combining factors in a precise manner to create a large prime gap, and that this
difficulty increases immensely as one moves from four digits to sixteen. Nonetheless, it is
hard to believe this difficulty increases so much as to make even 1 percent of possible
factors being used impossible with sixteen digits when all can be used in a large four-digit
gap is quite hard to believe.

Is there any way of working how hard precisely combining factors to create an unusually
large gap is??

--- In primenumbers@yahoogroups.com, "Andrey Kulsha" <Andrey_601@...> wrote:
>
> > The fact that the gap between 1327 and 1361 is the most persistent primal gap of all,
not
> > being equalled until between 8467 and 8501 makes me think of it as the most
unusual
> > prime gap that probably exists among numbers of any size.
>
> The "unusuality" could be measured dividing the gap length by log(p)*log(p)*sqrt(log
(log(p))), where p is the prime after the gap. Thus, the most "unusual" gaps (with
unusuality larger than 0.45) are:
>
> 6 between 23 and 29
> 14 between 113 and 127
> 34 between 1327 and 1361
> 1132 between 1693182318746371 and 1693182318747503
>
> I don't count the gap of 4 between 7 and 11, the gap of 2 between 3 and 5 and the
gap of 1 between 2 and 3, because log(log(p)) becomes inadmissibly small here.
>
> Note that the gap of 1132 is more unusual than the gap of 34, and, who knows, it
may appear the most unusual gap at all!
>
> P.S. The factorization of the product of all 1131 composites in this gap looks as
>
2^1127*3^568*5^285*7^189*11^112*13^94*17^70*19^63*23^51*29^40*31^37*37^3
1*41^27*43^27*47^24*53^23*59^20*61^19*67^18*71^16*73^15*79^15*83^15*89^1
4*97^12*101^11*103^11*107^11*109^10*113^10*127^9*131^9*137^9*139^8*149^7
*151^9*157^7*163^6*167^7*173^6*179^7*181^6*191^5*193^6*197^6*199^5*211^5*
223^5*227^5*229^5*233^5*239^4*241^4*251^4*257^5*263^5*269^4*271^5*277^5*
281^4*283^4*293^4*307^3*311^3*313^4*317^3*331^4*337^4*347^3*349^3*353^3*
359^3*367^3*373^3*379^3*383^3*389^3*397^3*401^3*409^3*419^3*421^2*431^2*
433^4*439^2*443^2*449^2*457^2*461^3*463^2*467^3*479^2*487^3*491^3*499^2*
503^2*509^2*521^2*523^2*541^2*547^2*557^2*563^2*569^2*571^2*577^2*587^2*
593^2*599^2*601^2*607^2*613^2*617^2*619^2*631*641^2*643^2*647^2*653^2*65
9^2*661*673^2*677*683^2*691^2*701*709^2*719*727^2*733^2*739^2*743^2*751*7
57^2*761^2*769*773^2*787^2*797^2*809*811^2*821^2*823^2*827^2*829^2*839*85
3*857^2*859^2*863*877^2*881*883*887^2*907*911*919*929*937*941*947*953*967*
971*977*983*991*997^2*1009*1013*1019*1021*1031*1033*1039*1049*1051*1061*1
063*1069*1087*1091*1093*1097*1103*1109*1117*1123*1129*1151*1153*1163*1171
*1181*1187*1193*1201*1217*1223*1229*1231*1237*1249*1259*1277*1279*1283*12
89*1291*1297*1301*1303*1307*1319*1321*1327*1367*1373*1381*1399*1423*1427*
1433*1439*1447*1451*1453*1471*1481*1483*1487*1489*1493*1499*1511*1543*154
9*1553*1567*1571*1583*1597*1601*1607*1609*1613*1619*1627*1637*1657*1667*1
669*1693*1699*1709*1733*1741*1753*1783*1787*1801*1811*1823*1861*1867*1871
*1877*1879*1889*1901*1907*1931*1933*1949*1973*1979*1987*1993*1997*1999*20
17*2027*2039*2069*2083*2111*2129*2131*2143*2153*2161*2213*2239*2243*2251*
2269*2273*2281*2287*2311*2333*2339*2341*2351*2357*2371*2383*2389*2411*241
7*2423*2447*2503*2521*2543*2551*2591*2593*2617*2621*2633*2647*2663*2671*2
683*2693*2729*2741*2797*2801*2833*2851*2917*2971*2999*3011*3041*3049*3067
*3119*3181*3187*3191*3221*3253*3299*3319*3331*3343*3371*3391*3407*3433*34
49*3457*3461*3467*3499*3533*3583*3607*3613*3617*3659*3671*3697*3719*3733*
3739*3769*3833*3847*3851*3889*3911*3919*4051*4091*4099*4127*4139*4177*425
3*4271*4327*4391*4421*4423*4441*4447*4457*4483*4567*4591*4721*4729*4787*4
793*4799*4877*4937*4967*4987*5021*5101*5153*5167*5189*5351*5417*5441*5479
*5483*5501*5563*5569*5573*5623*5647*5717*5737*5743*5801*5861*5881*5939*59
81*6089*6091*6199*6247*6277*6547*6571*6637*6679*6703*6709*6793*6841*6871*
6899*6907*6949*6983*7043*7109*7129*7151*7177*7229*7321*7369*7457*7517*757
7*7603*7757*8087*8117*8171*8387*8431*8467*8543*8669*8741*8837*8933*8969*8
999*9103*9323*9413*9463*9521*9613*9619*9697*9733*9829*9851*10139*10177*10
181*10337*10357*10501*10513*10529*10531*10567*10601*10657*10709*10781*10
889*10949*10973*10987*10993*11087*11113*11177*11273*11443*11813*11831*11
867*11959*11971*12269*12289*12409*12517*12539*12757*12823*12973*13009*13
457*13499*13523*13709*13721*14029*14153*14461*14717*14779*14783*14813*14
891*15139*15349*15401*15511*15649*15787*15809*16217*16267*16301*16607*16
649*16661*16747*16943*16963*17027*17029*17209*17393*17477*17807*17929*17
977*17987*18119*18257*18311*18379*19013*19139*19207*19531*20359*20719*20
731*20749*20771*20807*20921*21139*21157*21277*21649*21673*21787*22031*22
073*22193*22229*22571*22691*22769*22901*22961*23053*23291*23581*23599*23
869*23957*23981*24061*24229*24691*24749*24763*24793*25439*25463*25633*25
639*25703*26267*26699*26921*27011*27127*27281*27361*27527*27541*28607*28
879*29201*29387*29429*29527*29671*30941*31039*31159*31391*31393*31481*31
567*31573*31957*32359*32363*32843*33181*33329*33647*33829*34171*34429*34
607*34667*34703*34841*35053*35227*35449*35597*36241*36263*37273*37423*37
537*37657*37691*38069*38699*38917*39499*39901*40039*40127*40163*40993*41
333*41941*41947*42019*42187*42443*42989*43633*43997*44777*44893*45949*46
187*46273*47189*47563*47741*47791*48049*49603*50069*50849*51479*51787*52
021*52121*52391*53239*53401*54139*54293*54583*55259*56543*56747*57107*57
367*57373*57601*58789*59447*59699*61297*61657*62969*63029*63737*63803*63
977*64399*64601*66047*67021*67073*67139*68633*69767*71129*71429*71537*73
277*73571*76667*76943*77167*78059*78607*78643*79103*79579*80273*80557*82
031*83207*84053*84631*87281*87509*88589*89003*91807*92899*93307*93581*95
813*96797*97301*98947*99907*100559*101323*102217*102481*105251*105517*10
6163*110947*112799*113453*114761*115421*116491*118369*121013*121349*1276
81*128509*129263*129629*130003*130651*131101*132437*132851*133709*134081
*136403*136987*138727*139709*139987*141601*141629*146221*146347*148429*1
49249*151817*151841*152111*154073*154409*154681*158261*163393*165559*165
817*168353*172981*174257*175709*176159*176419*177623*177841*181607*18181
3*183809*185483*188189*194101*198337*199211*200689*203341*207029*211339*
215249*219937*222461*227191*228251*230611*233069*233419*238759*238943*24
2591*246707*246811*247697*251897*252877*254377*263647*265957*267593*2692
37*271501*276517*277021*277789*280037*283873*284819*292793*293473*295247
*296969*299191*299903*305867*308809*308851*311153*311447*316499*317857*3
19567*323413*323789*324953*329419*332567*336103*348709*353201*354791*361
469*362221*362419*364643*367949*371087*373463*374083*380929*386437*38797
7*388231*389629*389723*392599*404429*404693*405577*408137*425101*435649*
442789*452531*454039*454507*455269*474709*476023*478901*481681*486181*49
3657*495589*501503*504767*507077*509801*511831*514001*518239*521021*5267
39*530063*531989*558539*559067*564491*566701*572687*580711*581947*596419
*604729*632669*635519*640049*647963*687031*696523*698039*699191*701479*7
02817*705161*709901*727471*746797*749773*756853*761347*763027*764893*765
287*779231*780223*780347*794557*796921*808523*811501*813811*814199*82345
7*833737*843607*844867*844891*850567*869863*880283*909901*912047*931981*
939613*945341*951151*955613*967229*974657*976951*979177*984913*...
>
> [Non-text portions of this message have been removed]
>
• ... after ... all the primes below ... as I did in my ... divisible by 2, 3 or 5 and ... and that is as little as ... from a situation where ... Even with the
Message 3 of 6 , Feb 7, 2008
--- In primenumbers@yahoogroups.com, "julienbenney" <jpbenney@...> wrote:
>
> I have had a look at your factorisation of all the 1132 composites
after
> 1693182318746371 and it still does not seem as "efficient" in using
all the primes below
> the square root of that sixteen-digit number. Have you thought of,
as I did in my
> demonstration, of excluding all the numbers it that gap that are
divisible by 2, 3 or 5 and
> then comparing the factors?
>
> As it was I found 975 primes less than 975,000 occurring as factors,
and that is as little as
> 1.27 percent of all the primes over the duration. This is a far cry
from a situation where
> every possible prime less than the square root of a number occurs!
Even with the relativel
> persistent gap from 31397 to 31469 14 primes of 36 below the square
root of 31469 are
> required to divide all those numbers not divisible by 2, 3 or 5.
>
> I am really curious as to how much less probable it becomes that
every potential prime
> factor could occur over an unusually large prime gap. I do imagine
that there does exist a
> difficulty combining factors in a precise manner to create a large
prime gap, and that this
> difficulty increases immensely as one moves from four digits to
sixteen. Nonetheless, it is
> hard to believe this difficulty increases so much as to make even 1
percent of possible
> factors being used impossible with sixteen digits when all can be
used in a large four-digit
> gap is quite hard to believe.
>
> Is there any way of working how hard precisely combining factors to
create an unusually
> large gap is??

I'm supposing that there isn't such a way (now), or else we would have
a much better grip on how large gaps can be and where to find them.

Regarding the "efficiency" of prime factors in the gap, I noticed that
the gap of 13 composites between the primes 113 and 127 contains all
the primes up to and including 31.

The gap of 33 composites between the primes 1327 and 1361 contains all
the primes up to and including 43. This gap (I'm supposing) is the
last gap which contains all the primes up to the square root of where
the gap starts. So it *is* special. :)

Personally, I think the gap of 1131 composites between the primes
1693182318746371 and 1693182318747503 (which Jens mentioned) to be
the most outstanding, because it has the highest gap/(ln(p1))^2 ratio
(namely .9206).

The conventional "merit" scale of gap/ln(p1) seems to me to be more a
measure of the merit of those who are capable of searching high enough
to find such gaps, rather than of the gap itself! :)

Mark
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