20150910, 14:12  #133 
Nov 2007
Halifax, Nova Scotia
111000_{2} Posts 
These factorizations may be of interest to you:
8269^64 + 1 = 2 * 918268289 * 1044636673 * 27246569091713 * 330833653243776747964260771073 * 561890619505467417673578437249 * 22086806450819711302131630421735078367489 * 105934871592905257020705756143502134929049089 * 2292352902418162653788885412662352475559184256083067753620867701690641009153 460794822529^16 + 1 = 2 * 449 * 1153 * 224916338770049 * 5105921 * 389376737 * 233934808867765849950955026670045288476891302877629398472959953431198372257 * 38146602286961890605993948856974597290093265751212822436540098028242686596033 1136051159041^16 + 1 = 2 * 1889 * 9697 * 182657 * 1029233737591167061546404353 * 4991684400616824504643310216842312333983399021237982836112814552942824158689 * 223907903336764894448936105343784675277998190020105903822837625065223158354273 4219^64 + 1 = 2 * 641 * 1231149359617 * 359087389210482358273 * 47752688172237420031803545474157444702209 * 23905514849892476878667017966491475670464026031612378017268608727472758893569 * 15940971165884654293284337615436265671765632544321819539512208108813863512200961 1951^64 + 1 = 2 * 194823352251896321 * 5494903108627806024727424919107197441 * 202922654588164636965150039799151256762741981380536283306185198707673601 * 8679678916537243713117894814798421549692233950654818824374118595670136020932975203841 844734922753^16 + 1 = 2 * 11791364666078069125878775157281 * 42723450197711698022080547608002464275357153 * 66721653457086290019387247308738749397818644599571774538173279964206603452183917936744929089886575563256181290341377 381053332094977^16 + 1 = 2 * 97 * 8161 * 48193 * 2363393 * 7490371880343553 * 3561057220214818417811213269433920961 * 2205668062227165765587303425983768752328380602786776891201714602024637217 * 18624504053661432461756375938038441455140478620001566417771989642079451879522589698595877537 2437^64 + 1 = 2 * 257 * 50017651735988471295617 * 193033328300995328225784833 * 1653341336520595851600373108168771865756446751677205428506170372737 * 699314651919694529187619943356739714360111937971028210410414814371362999644527262590546130815248769 These are numbers that I helped my friend Karl Dilcher factor, for a recent paper John Cosgrave and he wrote together. John and Karl found all the smaller factors using a combination of Maple's ifactor command and GMPECM. (Maple's ifactor is a composite tool analogous to YAFU.) The largest two factors of each were found by me using GNFS via CADONFS. I posted these factorizations to factordb.com this morning. The size of the composite cofactors I split using GNFS were 120, 151, 154, 156, 157, 160, 164, and 166 digits, respectively. 
20150910, 15:31  #134 
Sep 2009
2^{6}·37 Posts 
Some of them look easier to factor with SNFS than with GNFS. Eg 2437^64+1 could be done as SNFS 217, which would certainly be easier than GNFS 166.
Chris 
20150910, 19:30  #135  
Nov 2007
Halifax, Nova Scotia
56_{10} Posts 
Quote:
Code:
3568837085257085726824324669332592915896684118818002242725852271403699379973925779342112031319428799393062121644906533510640858417589809041565976926535365031587312322325198092302028289 I've hit it with ECM up to 60 digits. 

20150910, 20:20  #136 
"Curtis"
Feb 2005
Riverside, CA
7×769 Posts 
I use 5/9(snfs difficulty) +30 to estimate equivalent GNFS difficulty. The heuristic produces 188 or 189, suggesting GNFS would be more efficient for this task. However, I've only used the conversion for SNFS projects in 200230 difficulty range, and don't know how well it extends to bigiron tasks.

20150910, 20:45  #137  
Sep 2008
Kansas
3×1,213 Posts 
Quote:
189 < 222 (remaining composite) which means SNFS would be quicker. Now, which poly would be the best? 13 * (13^51)^5 + 1 (13^43)^6 + 13^2 Edit: Never mind, FDB doesn't have all the factors. Last fiddled with by RichD on 20150910 at 20:47 

20150911, 02:13  #138  
"Curtis"
Feb 2005
Riverside, CA
7×769 Posts 
Quote:
I trusted OP's "184 digit cofactor" info. 

20150911, 02:40  #139 
Sep 2008
Kansas
3×1,213 Posts 

20150911, 15:38  #141 
Sep 2009
2^{6}·37 Posts 

20160304, 01:59  #142 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
23261_{8} Posts 
Two more for GF'_{3}:
Code:
GF(185,3) has a factor: 1784229339435*2^187+1 = 349993408840477174486123493730674197138441859438238753954983502151681 [TF:227:228:mmffgfn3 0.28 mfaktc_barrett236_F160_191gs] GF(208,3) has a factor: 4043684821205*2^209+1 = 3326950900831706864163191125935852440699815370218131646884454206899444776961 [TF:250:251:mmffgfn3 0.28 mfaktc_barrett252_F192_223gs] Last fiddled with by Batalov on 20160306 at 07:29 
20160504, 22:54  #143 
Apr 2012
993438: i1090
2·73 Posts 
Code:
463507*2^34186+1 is a Factor of GF(34185,6)!!!! 
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