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 2004-03-07, 14:33 #1 hbock     Feb 2003 16310 Posts Small range with high density of factors By accident I noticed a high occurrence of factors (59+60 bit) in the 37M range. 6 factors within 17 exponents or 12 factors within 155 exponents have been found (expected average is about 5 factors). Very often I've seen 2 or 3 factors close together (also during P-1 factoring) but never that much. I guess there are much more of such hot spots. OTH, there are big ranges with no factors found. Is there any mathematical theory available for that phenomenon? Code: M37636561 has a factor: 1135828231306295921 M37636591 no factor to 2^60, WZ1: 6947FA77 M37636609 no factor to 2^60, WZ1: 693CFA7E M37636619 has a factor: 511024828036639799 M37636633 has a factor: 570319236002859601 M37636673 no factor to 2^60, WZ1: 6942FA7C M37636699 no factor to 2^60, WZ1: 693FFA76 M37636847 has a factor: 631878724251180671 M37636897 no factor to 2^60, WZ1: 693AFA76 M37636913 no factor to 2^60, WZ1: 694AFA7B M37636981 no factor to 2^60, WZ1: 6955FA7D M37636999 has a factor: 597659352683030201 M37637053 no factor to 2^60, WZ1: 6946FA78 M37637087 no factor to 2^60, WZ1: 694BFA79 M37637137 no factor to 2^60, WZ1: 6943FA7F M37637227 no factor to 2^60, WZ1: 6946FA76 M37637287 has a factor: 405855766653864991 M37637359 no factor to 2^60, WZ1: 6957FA77 M37637377 no factor to 2^60, WZ1: 694CFA7E M37637381 no factor to 2^60, WZ1: 6950FA77 M37637419 no factor to 2^60, WZ1: 693CFA7C M37637437 no factor to 2^60, WZ1: 694EFA78 M37637491 no factor to 2^60, WZ1: 694AFA77 M37637519 no factor to 2^60, WZ1: 6949FA7D M37637521 no factor to 2^60, WZ1: 694BFA7F M37637557 no factor to 2^60, WZ1: 6952FA77 M37637573 no factor to 2^60, WZ1: 6945FA7C M37637581 has a factor: 407576741069557993 . . . M37638347 has a factor: 496922260335323759 . M37639141 has a factor: 692248891302635353 . M37640527 has a factor: 394094644116888527 . M37641883 has a factor: 1048689918573638239 . M37642147 has a factor: 522097453166506223 Another example (6 factors within 19 exponents tested) : Code: M37094417 has a factor: 345130813545920327 . M37095103 has a factor: 687749134153090337 M37095131 no factor to 2^60, WZ1: 6413F6E1 M37095193 no factor to 2^60, WZ1: 6417F6DD M37095199 no factor to 2^60, WZ1: 641DF6E3 M37095203 has a factor: 294807165076037863 M37095217 no factor to 2^60, WZ1: 6413F6DF M37095329 no factor to 2^60, WZ1: 640FF6E1 M37095403 no factor to 2^60, WZ1: 641FF6DE M37095451 no factor to 2^60, WZ1: 6415F6E2 M37095493 no factor to 2^60, WZ1: 6422F6E1 M37095517 no factor to 2^60, WZ1: 641DF6E3 M37095587 has a factor: 1105854523979714329 M37095593 has a factor: 401202455567851999 M37095841 no factor to 2^60, WZ1: 6423F6DD M37095889 has a factor: 680303185401949913 M37095893 no factor to 2^60, WZ1: 641DF6E5 M37095967 no factor to 2^60, WZ1: 6410F6E2 M37095977 no factor to 2^60, WZ1: 641AF6E1 M37096013 has a factor: 1014310337932612439 . M37096457 has a factor: 644196952627581481 Last fiddled with by hbock on 2004-03-07 at 14:35
2004-03-07, 19:51   #2
wblipp

"William"
May 2003
New Haven

2×32×131 Posts

Quote:
 Originally Posted by hbock 12 factors within 155 exponents have been found (expected average is about 5 factors) ... Is there any mathematical theory available for that phenomenon?
The expected frequency of clusters can be calculated from the Poisson Approximation. I attempted to describe this as part of the ElevenSmooth FAQ. The probability of finding at least 12 events in a space where, on average, you expect to find 5. is about 5 in a thousand. Counting the number of spaces is tricky because of overlaps, but if you divided the range into groups of 155 exponents before ever looking at them, you would expect about 1 in 200 ranges to have at least 12 factors.

William

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