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Old 2014-04-22, 21:07   #111
Batalov
 
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Phi(4,2^7658614+1)/2

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19238035476725*2^61+1 is a Factor of GF(60,12)
16909654924891*2^60+1 is a Factor of GF(59,5)
15365812010501*2^73+1 is a Factor of GF(72,5)

Last fiddled with by Batalov on 2014-04-25 at 06:15 Reason: + two more
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Old 2014-04-22, 23:52   #112
pinhodecarlos
 
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Quote:
Originally Posted by Batalov View Post
Yves shared a reference to the Calvo (2000) paper. Indeed, his theorem 2.1 explains the special case k|n and more.
Attaching the paper.
Attached Files
File Type: pdf A note on factors of generalized Fermat numbers.pdf (312.8 KB, 198 views)

Last fiddled with by pinhodecarlos on 2014-04-22 at 23:54
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Old 2014-04-23, 12:07   #113
pinhodecarlos
 
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This paper is free to be downloaded.
I opened at the university with access to sciencedirect.com and didn't notice it was free until I tried at home. Stupid of me!
Anyway, let me know if you guys need more papers from sciencedirect.com or other database. Just PM me.

Carlos
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Old 2014-06-07, 18:24   #114
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Phi(4,2^7658614+1)/2

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636417*2^35280+1 is a Factor of GF(35279,10)
477425*2^43389+1 is a Factor of GF(43386,10)
151059*2^51482+1 is a Factor of GF(51480,6)
99957*2^75859+1 is a Factor of GF(75858,5)
81935*2^78751+1 is a factor of GF(78749,10)
21379*2^121978+1 is a Factor of GF(121977,10)
26277*2^135854+1 is a Factor of GF(135852,8)

Last fiddled with by Batalov on 2014-06-11 at 16:27
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Old 2014-06-22, 17:20   #115
Batalov
 
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Phi(4,2^7658614+1)/2

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From mmff-gfn v.0.28:
Code:
370291543969*2^182+1 is a Factor of GF(181,3) [TF:220:221:mmff-gfn3 0.28 mfaktc_barrett224_F160_191gs]
407658847371*2^203+1 is a Factor of GF(199,5) [TF:241:242:mmff-gfn5 0.28 mfaktc_barrett247_F192_223gs]
15961621533*2^210+1 is a Factor of GF(206,12) [TF:243:244:mmff-gfn12 0.28 mfaktc_barrett247_F192_223gs]
GF(192,6) has a factor: 1044500744273*2^193+1 [TF:232:233:mmff-gfn6 0.28 mfaktc_barrett236_F192_223gs]

Last fiddled with by Batalov on 2014-06-24 at 02:49
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Old 2014-12-27, 01:09   #116
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Phi(4,2^7658614+1)/2

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49797593*2^8561+1 is a Factor of GF(8560,6) (for T.Nohara)
119002603*2^6988+1 is a Factor of GF(6986,5)

Last fiddled with by Batalov on 2014-12-27 at 19:32
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Old 2015-01-28, 06:02   #117
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No need to reserve a new range for the mmff-gfn6:

Code:
ECM found a factor in curve #5, stage #2
Sigma=1258624528355029, B1=11000000, B2=1100000000.
UID: ANONYMOUS, 6^8192+1 has a factor: 86958504245590708098930120015173020696577 (ECM curve 5, B1=11000000, B2=1100000000)
The count is a bit (but not much ) misleading .See the total counts for curves here: http://users.jyu.fi/~tamaraja/Fermat.html.

I added the factor to factordb and emailed Keller.
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Old 2015-01-28, 06:35   #118
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Oooh, this is such a nice find. I've been knocking at this one for a while too, but hey, -- hats off to you!
This is one of the very few that had not had known factors.
The cofactor, sadly, is still composite. ;-(
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Old 2015-03-14, 15:36   #119
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New large prime found during March PRPNet challenge:
27*2^5213635+1 is a Factor of xGF(5213634,7,6) and xGF(5213603,0,8)
http://primes.utm.edu/primes/page.php?id=119539

What does it mean when one of the bases is zero?
The first xGF is interpreted as 7^(2^5213634)+6^(2^5213634) which could be meaningful, but the second one I see as 0^(2^5213603)+8^(2^5213603) which makes no sense at all. (That number has no odd divisors, being a power of two.)
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Old 2015-03-14, 20:51   #120
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Quote:
Originally Posted by TheCount View Post
New large prime found during March PRPNet challenge:
Quote:
27*2^5213635+1 is a Factor of xGF(5213634,7,6) and xGF(5213603,0,8)
What does it mean when one of the bases is zero?
I guess that this could be a weird way of writing GF(5213603,8); recall that 0 to any positive power is 1, so xGF(5213603,0,8) would be the 82^5213603+1 cofactor (i.e. it doesn't divide F5213603 = 22^5213603+1, but the remainder after dividing it out, which is 22*5213603 - 22*5213603 + 1, because 8^x+1 is a sum of cubes).

However it is more likely that whoever ran the program ran into a bug (or a hardware error), because such large n-m difference are vanishingly rare: 5213635 >> 5213603. Someone has to re-run this with -a1 (e.g. like "pfgw -a1 -gos8 num" and "pfgw -a1 -gos2 num").

EDIT: and so it does; it divides GF(5213633,8)

Last fiddled with by Batalov on 2015-03-15 at 02:09
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Old 2015-03-15, 11:38   #121
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Quote:
Originally Posted by Batalov View Post
recall that 0 to any positive power is 1
¿Que?

ITYM "any postive integer to the power 0 is 1".
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