News - December 2008

• With P-1 (and small bounds for B1 and B2) I found these 313 factors of the series Phi_n(10).
  
  n=12027: 15491351886285623311.
  n=12239: 7273676603497351.
  n=12751: 2737494243171517.
  n=12771: 389641671664246441837.
  n=25820L: 23628827978516006261.
  n=26140L: 57754120777901.
  n=28340L: 57020097040558601.
  n=30100L: 16450031306801.
  n=30140M: 6020850671441.
  n=30220L: 2757550155307314120781.
  n=30220M: 545352415209001.
  n=30340L: 312357017976247201.
  n=30500L: 322301007703771001.
  n=30500L: 575895647203442846383501.
  n=30580M: 604898049464281.
  n=30700M: 64425021583501.
  n=31020L: 10247854149835501.
  n=31048: 50761554278849.
  n=31052: 21525710130569.
  n=31068: 704052622666621.
  n=31094: 47693462467099.
  n=31100L: 127167426198995101.
  n=31100M: 4522483158110101.
  n=31100M: 194873704839846701.
  n=31140L: 150670543442286061.
  n=31158: 2326855454914339.
  n=31188: 3099091939810353121.
  n=31234: 253667078986693.
  n=31242: 22649312734966899361.
  n=31260L: 83747878311707881.
  n=31278: 373629704058703.
  n=31312: 3889802493457.
  n=31340L: 1985477602222861.
  n=31392: 186402744963649.
  n=31398: 3631783051718209.
  n=31460L: 7659971564490961.
  n=31460L: 195398290682683661.
  n=31460L: 277804182261335635508461.
  n=31468: 11186311068949.
  n=31498: 47797775508407.
  n=31546: 7628981863133.
  n=31556: 759271147734989.
  n=31568: 19454167312784685560017.
  n=31640: 48039843503376481.
  n=31728: 31156806844321.
  n=31728: 260655392072017.
  n=31762: 22940522022860797.
  n=31788: 29497158998641.
  n=31788: 52904693967158269.
  n=31842: 4977471353306167.
  n=31902: 8218753901630299.
  n=31926: 3753706314091.
  n=31928: 3332318660867041.
  n=31968: 6433487336737.
  n=31988: 70581559617889.
  n=31990: 3628542877891.
  n=32000: 54789630784001.
  n=32020L: 60400639972721.
  n=32180L: 11545812847712682541.
  n=32220L: 45075303266758201.
  n=32300L: 468800497832216101.
  n=32300M: 316394405779002158501.
  n=32460M: 12981698099343702538801.
  n=32500L: 27178637055752812501.
  n=32580M: 10589226338521.
  n=32620M: 225063784171381.
  n=32780L: 39256581788770061.
  n=32860L: 4384911732314741.
  n=32900M: 19686893891125301.
  n=32900M: 4026802484800086601.
  n=32980L: 4704879395570381.
  n=32980M: 7890055738565578741.
  n=33340L: 47621223773621.
  n=33580L: 25142368647838501.
  n=33620M: 31126330333421.
  n=33740M: 192735823305061.
  n=33740M: 479650991386050988001.
  n=36100L: 105735915950101.
  n=36100M: 70154930668001.
  n=36140M: 44342816256896821.
  n=36260M: 63240518644488355801.
  n=36380L: 161620358193241.
  n=36460L: 392464374068018321581.
  n=36500L: 490838050941001.
  n=36500L: 106560914522263501.
  n=36700L: 269358996886801.
  n=36700M: 58737087373201.
  n=36780L: 861455839331641.
  n=36820L: 87530766778880741.
  n=36980L: 16096174354780241.
  n=36980M: 13735016968438861.
  n=40100L: 501778718299501.
  n=40140M: 8623174485241.
  n=40180M: 1243677060576529181.
  n=40220M: 415141197817959341.
  n=40300M: 131817927816901.
  n=40500L: 15643457910001.
  n=40500M: 309564082516501.
  n=40780L: 358884393547861.
  n=40820L: 5803632955348541.
  n=40820M: 1635038934139001.
  n=40820M: 55780798881135901.
  n=40980M: 285936437160764521.
  n=41020L: 39357512566719341.
  n=41380L: 15872211097961.
  n=41460L: 25437570268741.
  n=41500L: 25460718618001.
  n=41660M: 195560331423161.
  n=42020L: 1169085620897921.
  n=42380M: 66632890326601.
  n=43060L: 2342729171015501.
  n=43140L: 311177419139341.
  n=43140L: 675512996349421.
  n=43180M: 50020221351281.
  n=43220M: 10090673242262681.
  n=43380M: 532414493550721.
  n=43420L: 4293752086781.
  n=43500L: 4225539141366001.
  n=43620M: 4602035970328861.
  n=43900L: 20560168670801.
  n=43900M: 904620152459501.
  n=44140M: 892895704596041.
  n=44220L: 903878696417401.
  n=44380L: 20708126414641.
  n=44380M: 86136865171984820127481.
  n=44420M: 4576337096161.
  n=44540M: 22664631427031261.
  n=44660L: 6572492539291141.
  n=44660M: 6980942037800501.
  n=44940M: 9676217271841.
  n=45100L: 33313368732601.
  n=45380M: 5749782911461.
  n=45420M: 377523728152141.
  n=45460L: 94723532960101.
  n=45500L: 208892345434501.
  n=45580L: 279708092774064961.
  n=45620M: 18595746707221.
  n=45620M: 2109277806213121.
  n=45700L: 17851222093130401.
  n=45700M: 108173180514001.
  n=46060L: 2880675233474921.
  n=46100M: 22684958853210601.
  n=46100M: 29535248060220382201.
  n=46140L: 3911500476662070121.
  n=46180M: 198046925124161.
  n=46260M: 48891473824321.
  n=46340L: 14130263282826461.
  n=46380M: 15094913553241.
  n=46380M: 1199055428960421661.
  n=46460L: 12599294358701.
  n=46540L: 30974424942332041.
  n=46620L: 12479183625803895001.
  n=46700L: 3165199723201.
  n=46740M: 473927294608158928956361.
  n=46780L: 143636204168862651821.
  n=46820L: 1692346179863161.
  n=46860L: 7901678794021.
  n=46860L: 100948469899201.
  n=46940M: 8168612911141.
  n=47060M: 33103717078121.
  n=47060M: 743769081715141.
  n=47340L: 25842331529101.
  n=47380L: 3705406945465781.
  n=47420L: 3952037664941.
  n=47420L: 21341233197481.
  n=47500M: 11354352690001.
  n=47500M: 57262433842501.
  n=47580L: 1014844460838781.
  n=47900M: 2314676036339101.
  n=47900M: 298177505970974501.
  n=47940L: 4995896673301.
  n=48380L: 3823368834401.
  n=48420L: 17239436608861.
  n=49060M: 477524102959201.
  n=49180L: 61155720833461.
  n=49180M: 6980195350301.
  n=49260M: 3595500250357681.
  n=49300L: 322542763505201.
  n=49300M: 220041670034701.
  n=49340M: 15009333192881.
  n=49420L: 21527548049141.
  n=49580L: 800696741810321.
  n=50180M: 1848350345299901.
  n=50700L: 48748789155301.
  n=50860L: 28718579728721.
  n=51180M: 127194600857101.
  n=51460M: 236886350456621.
  n=51980M: 3407790191161.
  n=52020L: 15177495081781.
  n=52020L: 71971118377056948241.
  n=52500L: 3301708252501.
  n=53460L: 505463514298381.
  n=53860L: 6469510560761.
  n=55020M: 16333901291401.
  n=56300L: 140277278063401.
  n=56460L: 74415297465661.
  n=56700L: 3308438479501.
  n=56740L: 569928111979201.
  n=57900M: 58204164622501.
  n=58100L: 314434982148701.
  n=58580L: 569667155225621.
  n=59580L: 730602142971121.
  n=59660L: 5544050414141.
  n=59660M: 82426497145721.
  n=60100M: 1662695220227401.
  n=61900M: 1334228833474501.
  n=62900M: 13429579732801.
  n=63380M: 161292901343701.
  n=63580M: 2848845353074381.
  n=63620M: 5051385247361.
  n=63820L: 18307119158761.
  n=63980L: 530506959887521.
  n=66019: 7646801198321.
  n=66180L: 14278815202081.
  n=67300L: 88078686290801.
  n=67340M: 594971728261.
  n=67380L: 1484259981241.
  n=67580M: 50757364799161.
  n=67900L: 1451273577481001.
  n=67900M: 6413955744701.
  n=67940M: 8887033558721.
  n=68220M: 1020848334676741.
  n=68340L: 6533976670621.
  n=68420M: 7712212427701.
  n=68500L: 688468429001.
  n=68540M: 566112092981.
  n=68580L: 598780278181.
  n=68620L: 2470208839772027381.
  n=68620M: 3013842345341.
  n=68780L: 13075174429561.
  n=68780M: 480159782881.
  n=68820L: 2520539657281.
  n=68820M: 37884528415801.
  n=68860L: 1686095699861.
  n=68860M: 109338594641761.
  n=68900L: 16628093893262801.
  n=68900M: 93539064079501.
  n=68980L: 93794432005601.
  n=70020M: 2278729969741.
  n=70100L: 2140731325001.
  n=70220M: 9137998105202641.
  n=70700M: 1997358733898501.
  n=70740M: 20876661468001.
  n=71300L: 6393763142195801.
  n=71300M: 2335593778801.
  n=72020M: 2531077793921.
  n=72060L: 64152254643121.
  n=72060L: 151291269530041.
  n=72060L: 396655564214534101.
  n=72140L: 47686786540596001.
  n=72260L: 1805309371981.
  n=72300L: 575703571501.
  n=72300M: 620197503456301.
  n=72420M: 585760464135522204181.
  n=72460M: 613539181261.
  n=72460M: 1584052407601.
  n=72500L: 250058982627302501.
  n=72540L: 55307884813585492141.
  n=72580L: 637352415241.
  n=72580L: 15918307121955649441.
  n=72620L: 33023688506161.
  n=72660L: 3614103241141.
  n=72660M: 10211491661281.
  n=72700L: 3118935851201.
  n=72820L: 3352349603021.
  n=72820M: 294124399809901.
  n=72900M: 918603787501.
  n=72900M: 1029793929301.
  n=72940M: 3570030575581.
  n=72980L: 69157180827479045801.
  n=73500L: 2793132079501.
  n=74700M: 21591922277701.
  n=76540L: 8503863726961.
  n=78140L: 419748566578907981.
  n=78140M: 65300197418641.
  n=78420L: 2083998412212043681.
  n=78500M: 531688550567501.
  n=78540L: 4351521109321.
  n=78660M: 10826799910830181.
  n=78700L: 619538519801.
  n=78740L: 259047850357200101.
  n=78820M: 4402618709581.
  n=78860L: 5838601823881.
  n=79060L: 230111362250681.
  n=79140L: 803092855861.
  n=79860M: 15600479369519881.
  n=79900L: 46910243366801.
  n=79980M: 105817869477361.
  n=80220M: 6074343032101.
  n=82260M: 756255335761.
  n=85140M: 22014009121501.
  n=85580M: 987848142821.
  n=86020L: 481709247361.
  n=86060M: 1850990270221.
  n=86460M: 8459981137081.
  n=86460M: 18489638300101.
  n=86500M: 539943985501.
  n=86860M: 1974386777941.
  n=87620M: 2965156130561.
  n=88100L: 660705069001.
  n=88340L: 10005333629201.
  n=89300M: 4831747133814901.
  n=90620M: 1190319436081.
  n=92500M: 67827880705001.
  n=92780M: 2054091861961.
  n=99100L: 1142888626252203601.
  n=99260L: 74089319466949963961.
  n=99380L: 174161278701237761.
  n=99484: 2867491710583501.
  n=99540M: 49492277341995601.
  n=99858: 129290805583369.
  n=99940M: 20546048623807601.
  n=99940M: 17594820928226575015141.
  
  No factor of Phi_67900L(10), Phi_67940M(10) or Phi_70700M(10) was known yet.
  The factor of Phi_66019(10) shows that x65296_9000000000_3344269890 was composite.
  
• With P-1 and somewhat larger bounds (B1=1e6 and B2=1e9), these 19 factors were found.
  
  n=24100M: 581599948687698673801.
  n=24180M: 8893457355802485643321.
  n=24220M: 128522329396707215341.
  n=24296: 3217754783086657439273.
  n=24322: 316665666809441.
  n=24326: 133163692288445999196343.
  n=24298: 102557490408413570009.
  n=24316: 3438811078350421.
  n=24330: 1420327009557121.
  n=24346: 1103427716491409.
  n=24364: 11009432265330483641.
  n=24366: 113191665402726007.
  n=24368: 37152417426213937.
  n=24370: 529328797354463090441.
  n=24500L: 15899383015992001.
  n=24700M: 18680063431360501.
  n=24940M: 54330077171079041.
  n=24980M: 11891252958812861.
  n=24980M: 54509588302586712761.
  
  
• n=23300L: 33254549797154507201.
  With this factor I try to restart my search on numbers of the form 10^n+1 (and n<12001).
  
• On Dec 01, 2008, I asserted the compositeness of 11 numbers - not the best of my ideas.
  Here I list 11 factors, which prove the assertion.
  
  n=66009: 214118399627055883.
  n=99122: 519071979157.
  n=99152: 22724685648433.
  n=99325: 487031794201.
  n=99392: 184935487119553.
  n=99435: 1446698111041.
  n=99496: 8268153219569.
  n=99665: 3632889712321.
  n=99704: 58865433031681.
  n=99728: 197811448979009.
  n=99818: 2727989905703.
  
• The following factors were found with p-1 (B1=1e6 and B2=1e9).
  
  n=29020M: 244879901073293061198041.
  n=29580L: 18558860975512066081.
  n=29620M: 154456985552306968502861.