20061115, 12:21  #1 
Dec 2005
2^{2}×7^{2} Posts 
Small factors
I was looking at Primenet results for cleared exponents. As I looked at the largest factors found (size 103 bits), I was surprised that regularly, these factors were not prime but contained very small factors. I thought that all exponents available were at least trialfactored upto a certain level. So how can it be that factors like 23 (for exponent 36773851) were missed ? Or are some people just assigning themselves exponents without bothering to check whether they are trialfactored ? I guess I am missing something but would appreciate a clarification.

20061115, 13:26  #2  
Jun 2003
2^{3}×607 Posts 
Quote:
Last fiddled with by axn on 20061115 at 13:26 

20061115, 13:52  #3 
Dec 2005
304_{8} Posts 
and I suppose it also truncates the "bit"length ? The factor I am talking about is indicated as
103 9075527700594867141608327604401 taking the log clearly indicates that 103 is the correct bitlength of this number, so how can it be truncated (which I understand as being chopped at a certain point in the sequence) ? The above number has the factorisation 23*239*6709*55313*163861*27150982078609 where the first five factors are all smaller than 2^18 
20061115, 19:41  #4 
Jun 2003
2^{3}·607 Posts 
Apparently so
Here are a bunch of factors truncated in the report to 31 digits. Code:
32492333 103 F 8316861747465793506084499558121 09Nov06 19:21 cathas CE8CFA671 36411527 103 F 7263852526156696381869159290527 11Nov06 22:30 blackguard carbon 36773851 103 F 9075527700594867141608327604401 14Nov06 17:53 S517661 C7F0535E6 36534737 101 F 3109119109442520160313833481551 11Nov06 13:06 S152209 CFC460636 36626063 101 F 1875630778194861452245225486337 01Nov06 09:11 abienvenu betaweb1 36627907 100 F 1746551189471568749237051498287 03Nov06 20:11 mnrcrl42 silvia PS: The truncated factors are clearly not valid, since a factor of 2^p1 must be of the form 2kp+1. So the smallest possible factor is 2p+1. If you see anything smaller, obviously it is not correct. 
20061115, 19:55  #5 
Jun 2003
2^{3}×607 Posts 
Well three of them are valid
Code:
8316861747465793506084499558121 = 37 * 1097101 * 204885463807621385719433 7263852526156696381869159290527 = 7263852526156696381869159290527 9075527700594867141608327604401 = 23 * 239 * 6709 * 55313 * 163861 * 27150982078609 3109119109442520160313833481551 = 127 * 919 * 26639012872966337600043127 1875630778194861452245225486337 = 1875630778194861452245225486337 1746551189471568749237051498287 = 1746551189471568749237051498287 
20061115, 23:34  #6 
Mar 2005
Internet; Ukraine, Kiev
110010111_{2} Posts 
If you need to get full factors, you can try guessing the last one or two digits and test if it is the real factor  only 50 odd numbers to try. This is simple with help of a little program (no, I don't have such a program).

20061116, 00:12  #7 
"Mark"
Feb 2003
Sydney
3·191 Posts 
Three of them are in the latest factor.cmp, and their (prime!) factors are:
Code:
32492333,38316861747465793506084499558121 36626063,1875630778194861452245225486337 36627907,1746551189471568749237051498287 
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