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.

:cat: :newcat:

[QUOTE=Kees;91573]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.
[/QUOTE]

The primenet report truncates the really long factors.

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

:cat: :newcat:

[QUOTE=Kees;91581]and I suppose it also truncates the "bit"-length ? [/QUOTE]

Apparently so :sad:

Here are a bunch of factors truncated in the report to 31 digits.
[code]
32492333 103 F 8316861747465793506084499558121 09-Nov-06 19:21 cathas CE8CFA671
36411527 103 F 7263852526156696381869159290527 11-Nov-06 22:30 blackguard carbon
36773851 103 F 9075527700594867141608327604401 14-Nov-06 17:53 S517661 C7F0535E6
36534737 101 F 3109119109442520160313833481551 11-Nov-06 13:06 S152209 CFC460636
36626063 101 F 1875630778194861452245225486337 01-Nov-06 09:11 abienvenu betaweb1
36627907 100 F 1746551189471568749237051498287 03-Nov-06 20:11 mnrcrl42 silvia
[/code]

What happens is sometimes P-1 finds two factors in a single run and reports it as a huge composite. The report truncates them, but the DB has the actual value. So I guess you could ask George to give you the real values:wink:

PS:- The truncated factors are clearly not valid, since a factor of 2^p-1 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.

Well three of them are valid :smile:
[code]
8316861747465793506084499558121 = 37 * 1097101 * 204885463807621385719433
7263852526156696381869159290527 = 7263852526156696381869159290527
9075527700594867141608327604401 = 23 * 239 * 6709 * 55313 * 163861 * 27150982078609
3109119109442520160313833481551 = 127 * 919 * 26639012872966337600043127
1875630778194861452245225486337 = 1875630778194861452245225486337
1746551189471568749237051498287 = 1746551189471568749237051498287
[/code]

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).

Three of them are in the latest factor.cmp, and their (prime!) factors are:
[CODE]32492333,38316861747465793506084499558121
36626063,1875630778194861452245225486337
36627907,1746551189471568749237051498287[/CODE]The report truncates the first digit/s.