Largest Consecutive Factorizations

Introduction

News

Records

Factorizations

k=2: 24862048 digits.
     n   = 2^82589933-1 (Mersenne prime)
     n+1 = 2^82589933

k=3: 388342 digits.
    (n   = 2618163402417*2^1290001-2)
     n   = 2*(2618163402417*2^1290000-1) (2*prime)
     n+1 = 2618163402417*2^1290001-1 (prime)
     n+2 = 2^1290001*3^2*290907044713

k=4: 12202 digits.
    (n   = 1001056355*2^40504)
     n   = 2^40504*5*13*15400867
     n+1 = 3*11*17*9743*2723504659773219359123*p12174
     n+2 = 2*(1001056355*2^40503+1) (2*prime)
     n+3 = 1001056355*2^40504+3 (prime)

     Primo certificate for p12174 is in the Download link under "Primality" at 
     http://www.factordb.com/index.php?id=1100000000777825386.

k=5: 3414 digits.
    (n   = 1413937732*7937#-1)
     n   = 579065494045723757809670167419583*p3381
     n+1 = 2^2*397*521*1709*7937#
     n+2 = 1413937732*7937#+1 (prime)
     n+3 = 2*(706968866*7937#+1) (2*prime)
     n+4 = 3*(1413937732*7937#/3+1) (3*prime)

     Primo certificate for p3381 is in
     http://primerecords.dk/certif/k5dig3414.zip.

k=6: 2971 digits.
    (n   = 36133626794*6907#*4-4)
     n   = 2*2*p2970
     n+1 = 3*82988833*p2963
     n+2 = 2*p2971
     n+3 = 36133626794*6907#*4-1 (twin prime)
     n+4 = 2^3*18066813397*6907#
     n+5 = 36133626794*6907#*4+1 (twin prime)

     n-1 = 5*41479*9482120399*692307937*131322060679698527*181584030482638864427*91941573397640066926253*590631941644223822689303*c2863
     n+6 = 2*29202485169059*8174320284005081474951269*c2932

     Primo certificate for p2963 is in the Download link under "Primality" at 
     http://factordb.com/index.php?id=1100000000935801201.

k=7: 1493 digits.
    (n   = 410797845232*3499#-6)
     n   = 2*3^2*173273*1166749799457452217869285429*p1459
     n+1 = 5*2385113*p1486
     n+2 = 2^2*(102699461308*3499#-1) (2^2*prime)
     n+3 = 3*12418053157*305955714712481*893413432297756873*p1450
     n+4 = 2*(205398922616*3499#-1) (2*prime)
     n+5 = 410797845232*3499#-1 (prime)
     n+6 = 2^4*3499#*25674865327

     Primo certificates for the non-trivial proofs p1459, p1486 and p1450 are in
     http://primerecords.dk/certif/k7dig1493.zip.

k=8: 850 digits.
    (n   = 72*(y-2^2)*(y-2^8)*(y-5^4)*(y-21^2)/14!-3, 
           where x=429*(2^50+9096237) and y=(16*x^6-72*x^4+81*x^2-25)^2)
     n   = 3*p849
     n+1 = 2*9277*9862129*3520556383*110169881898372133*p812
     n+2 = p850a
     n+3 = 2^14*3^12*5^2*11*13*17^2*71*97*127*151*163^2*191^2*463*857*911*2039^2*3121^2*3671^2*27617*94559*105397*334363*818093*1081361*1094881*8032223*8673499*28379557*379425797*1928145619*33487856029*75107484533*1932425260981*9461343831419^2*14124202973927*43900119421747^2*78263386849913*87716528455057^2*138911112511313*332653160927821*4685494971602672527*22578112014348327386927*122263255338736449624461*53567894301100706956015487*55113771707297541288419321*255922537221444436661056643^2*145432951087573335616229844260519*4900032397637694849377097976316041*926379724905047662387400079983152469*3535157730088988059299768749292371183*55948241477020996875929924604390118463*3743940973665875353073451608837014391941255351681351250046220217081500959197080629383389*203171345168977554057804535252209695979589563860186352662241349515657691011371586515901396031719887008989179
     n+4 = 7*23^3*37^2*79*83*131*167*257*359*1399*2269*3931*5113*7523*13103*89101*132299*213281*263089*851549*4249391*6665843*111758329*539610871*752205743*1174151543*31955728957*36525034891*69815079791*268077520427*1691563012343*2886922982807*4975707238891*59360469637561*212588265532079*5113751487906046691*8073995731340531749*85132972807816646399*698440825237695963167*235466779559351867156089*8638974032935885408356031*20630841274038888637908941*23544276279027077345208343691*65273164625411946236961729023*2805682321785697112266888628881*7463890134212550090674435831341*301121597780546403842070623578537*5865153635770685042150185867353407929*6528501668900596816131526302366696161*297583971826860297080110478721751715863*609922395430796703126238760038151517372382559*11490578631615430896997480933928606585904494641396939747239142030295759*544384709264090687490313199283910398850735650743667894915780898418916503024247696440699877047
     n+5 = 2*p850b
     n+6 = 3*19*1009*p845
     n+7 = 2^2*208577*p844
     p850a and p850b are different 850-digit primes.

     n-1 = 2^2*95819*27609121*c837
     n+8 = 5*311137*114685080442874011127*c824
     n-1 and n+8 are unlikely to have other prime factors below 10^25.

     There is a checked verification for k=8 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifacgast.zip.

k=9: 804 digits.
    (n   = (y-23^2)*(y-24^2)/55440-8, where x=2310*(10^30+40790547)+5, t=5*x^3-x^2-x-1, y=(t*(5*t+9)/2-31)^2)
     n   = 2^2*4261*75337*110917*574283*15755933*255946099*166880080123*p757
     n+1 = 3^3*5*331*p799
     n+2 = 2*61*2011*10601*444546250219*p783
     n+3 = 2423*101557755913*266403633889*p778
     n+4 = 2^11*3*13*19*23^2*43*47*53*71*79*103*113*131*313*389*619*1093*1579*7451*13309*33569*76757*92627*1319261*1114238843*18290398183*489625035961*1270641752531*52875541480393*3354903562911899*52004118381270427*63033107717874313*717795020292627461*23659571581081290354933869*3702831616477791607996624583*257083647123710228163986163068933*72995580604330472767675305956448577*5650656063178953440230244461478711321*103254525137730884356325462161948293611*32639399163965702665322588630095762611168465302871817*290963797629785789829587858554435806099102183324391210019*344388126566808701830834392491420283719805371748824281502279*25680373978698894578006006245685748564269008939777185093119975580853*7423574759509261460819979768473950225646324801051680510010128954872507181358849536795703*15516753433189944918910992099992845607888494478960888121798680997688704717693550507172185714219
     n+5 = 7*17*37*157*283*317*419*691*1433*1901*1913*6689*2439953*2653993787*3079307041*6361306879*1848987653587*147279592197387043*1289593158350311282393141*258152730541656904182113569565579770280957857*1890907634926683857961173716314441547212123672350629*1159459696029019298615512562569201052337715269676992150571*101395877354331666235965890216235926227140750818573375760797579*4799995106140499339265636929746263616111770974981245944687585577*61795841127498738668772363572553955238962417043039906825237172011091635954239597*5715444996056603421288522870733528787435898718545067670986221446315823787588705707749193186082986329937601489837900897917020845211645033463227*54076763599622413525359748434676447007280788635568722994736593287262429063324143776398234293027462569762217074967207645461719612421989698929945491374185704164228447081879263525107032319
     n+6 = 2*5*p803a
     n+7 = 3*p803b
     n+8 = 2^2*11^2*29*73*643*727*3547*86851*915283*4719726079*14266216717*19750153781*172788128959*17346192250609*365172908112440801*11349953878687101971*2561508111902244874441*79444083751332871206115014253*450306550911446554674008453280965308352893868510532064297*263835423801369863013730916110752495832191782138880338638851594918414481883131465152263052713273663*61631955000000000000007542003471783026400000307642447106035732572941622967899510259568649113820727673*3649682510725537007425276195322911915332208177782676302046766103340728769618932807593262154741181712973564003103*329717977532262240764745389133388989475063541661399274000306678512865674777976547994209353911413438933330737758598303758122047*36666734407117219300514766129735427723031863623789410226119966507111395237732790320214641435454433341255486268836866249362203194571981532931307
     p803a and p803b are different 803-digit primes.

     n-1 = 41*535101467*c793
     n+9 = 599*919*3779*42649*16228241*61486981526111*c769
     ~2000 ECM curves with B1=1000000 and ~500 with B1=3000000 

     There is a checked verification for k=9 in 
     http://immortaltheory.com/cnt/VERIFY_804.gp.

k=10: 552 digits. Same as n+1 to n+10 in record for k=11.

k=11: 552 digits.
    (n   = (2*x^3-3*x^2-5*x+2)^2*(4*x^6-12*x^5-11*x^4+38*x^3+13*x^2-20*x-6)^2/16-9, where x=15^26+10620028)
     n   = 3^3*7*11*29*797*1039*1103*1427*3251*3259*39103*204013*535169*272886083*4121448017*21598899661*409072561753*519537015073*858143907144487*2678465528135027*692182816348603133*439911034040804233169*153715102416475789067816753*54797332294839573582156690577*440167453052484845395296777659759700025866945940187*1490753961804495511472010679553553719994851286296636790921170578750322611815602909990587172619578278369031028762662879076160944273007*1450680572742737082125066423130337717154786020427056296586957684578732972366536675987593477013262565042583501426333299494101797785885285518865883802022347
     n+1 = 2^3*43487*34434429143*2671318275315433*144391571132721412007659975773678785330610089905539137*940513682365964521411580467479612497184247667346713718331165504974197162360066016886090000157831159486454280417507615487658913*23697051431951006121549913068636179160276931840635795389569544502842632838052313094193084394909824275746194884055203476348365691538147573536676825410145438263097877759897321813731363102528352135040888060614361490454942144750332454109731385394101534920099785563323780503177717538198321927198981106741255758046649749837975685343085297371886447
     n+2 = 55337*p547
     n+3 = 2*3*5*47*331*359*1291*986981*2002953749*12712144849384397587542162535943365250597239073952624539389402782482235601428590827634430510713300948536358810305637766909036928823755797859532604601090124387614285823*18945361041423043311319369017577973246669307283626167890232229413293367285951608091227742359201034685435855643761232140422680237863456889876666739738438100498603887595881151447426043074385956576132662399924073599216085551343603013723886900327594151459667641630117816913419048703647696736246088622541342174916158369771383055067726471952087324924256509511261829
     n+4 = 31*89*181*6361*p542
     n+5 = 2^2*17*1613*3607*69127*424841*425109049*6325447759*10312284591953317*746927965061759029*19767145144099170403669*123791177180274476752043071436801231*2952042386335628044687126353949453187623*178481957758622318842860130819690665371964998256863*300927303212925356381171755013770189743897160452800095868728344049290879607763303843462053162212559*1102981670154685461815124635819650935110418538347527302950473117871569767406358091115131099626671177611316414412260909513756090628405728547752055120012571638940984062942925356854780462161945601138704897404375003740810066511414451418641
     n+6 = 3*457*6781*13042499*215026223*2776982627*945861378949*25483746753600361*1225228327490491722399383*3048651749705940194612034637*98314563435243698633253261744130446808398230218091*1246623054122463570492075535372545413205158962595747661882605745363469309593103195271046193*1581035282528170936207659753899395448814090750468632122640146026524635806886917078047742443962934242556072610589811768073747816162973*81527183254529280633866482960793949729652462638349992191744411300813558915055882521129101344915359014881870035032987964636684423048773673044373977789429425279536633077
     n+7 = 2*7*199*383*22769*1981631*228901271*7062362423*9533727384721*906067834230707143*64730151242356234799*107292386616952796862417353040542655607*991359526871128739001255602307265276247*833250990595522848394855160380170407624056328390633*118542861243684331346133060578940628788108164945643764750449504847*313137269528035784179493718338224423165522666921145657071280783527794991*53128518216684001749484129438868239369239300198595888609347503993013015448440073685348575297*13538811187409942704866045458529013717912051438213362623493323310488245254431519469209178056918451750559268479
     n+8 = 5^2*13*109*157*179*10354103*461033426457284890164569*1074664477013774541471015856723*236379140653122682533795791912035915737436342452064899707*836594247678316140409926696591950566029481721475866669170339474308266771*19718256313056761286183243497093070095538402382589116249875457477348917609808894542321259874568820933871344039*5171259921575516811118212485227773865963786606652017136230584806842149049773673503246391413166744435127670198554010570357245742073008066896203908314286152240747748816381591037006910281196887913693451767471640477347962041531413024457148719868747
     n+9 = 2^8*3^2*41^2*103^2*27982299736623709616597^2*7594909640655948575910427^2*17674809061335434031710009^2*110187914144408232433740027965203^2*4341485828238153721926363725581225345540759229754953224439159767^2*881026963019664248702772688343335649876074773037498980427802485251903777999983143151021705494380819239^2
     n+10= 1061*1877*p545

     n-1 = 2*14723*c547
     n+11= 2*11^2*19*1884523*5051815528104032427515047480883*19320078030071947802324076404344843659439811*c468
     ~2000 ECM curves with B1=1000000

     There is a checked verification for k=11 in 
     http://immortaltheory.com/cnt/verify_k11_c552.gp.

k=12: 521 digits. 
    (n   = y*(2*y-5)^2-9, where y=(x^3-13*x-4)^2/64 and x=2^97+51514439)
     n   = 2^5*3^2*7*13*17*887*2437*13339*30089*49787*295847*699089*631106473*11758433179*73686673513*7997685879317*360477880611768553*518091460634382709*15050964904693165182299*17917308789525613965551*1171398140920239420717280663859*5728960191355159203801240850367037084832744087*92392243208124716402416696390826938854637442681347332860841289*121236226055886382351608060819534752703322469026614011168245516475418263183019798929549185071705234086761*918349743607285428786924077404455720960195812218071791254827049991798749780311888599251213291805037642077080458866120179
     n+1 = 257*23296519*1242096601*3048355236167*67973327977858879*135996879951639809*606635802341662435132175179349293714549272051430455678036875643598915349310782211721973275723042298531808064519*476049241189471357760864128686221766350096529085909228324245109554467660977093847926313967394428260917981014913568279257549965254742677328761804745011575095142481178743211265579508624993461874944839192347838538543548727181449071950041770613073355148416139112738412832441110967186585544024820400112942605725121208812404616109968539147733573432097
     n+2 = 2*6857*p517
     n+3 = 3*5*43*863*1747*3581*21019*179951*2665988320602598119712037148890597589508913576108560151252151200435666518468042695203986763274362208546291758588761326187257881915092235910607165312317946140409466980679*1723456019694535985024052831278769982409850693731036183019401249854005649027058715170650019790949774813382424909094653917094387303200560837954607615294370400024910950769859150808458713206581975301028261726486069902159812186060487965885791282409908342513319078236674752345844445787295726398459700590072492402991653840119101403324069
     n+4 = 2^2*59*22091*35999*p510
     n+5 = 11*29*101*571*3691*5419*65203*1295389*2920079*138203671*6079815041*6807805279087*4214598470855783*252775275481837740704983146954700963713586730153850922755617*576545967736398377880005536912745355844295932795061127507304915071378800145932900134220140776910446588573801936420449624688244935867703659255654133*189791581562086004182786053911463893927100346909312552275180515756725069187771439438525215323275001305235375297858441337994282392220999296010244121821682564249311868864678056128116961654457526895927237549982385731864650276028768837996759
     n+6 = 2*3*54881331191*3549757051333*112712844601859*345742433728700149*61347189073381493207918536815318678415683328792843643*14608757911619482325513417398922661995740823683847143154630043302377343949721739*2384710061042407663350652645312256059993060891913655975778396102465978591303335049*48974153515499894645234965703965797304135752880534893557058100045319995640327414406010484894887092334290124777*12693500454889611206532256075665637940212171162010189388440729662267877275515941250815914080365061254677361638056776706610356922013384369276221
     n+7 = 7*23*47*1823*2273*2447*2801*5517624133009231*298951889844775159*1583866351585911133838401*62186682346004723275580488729752311*1468062138781954877605806181091162927*17659000145579387114655129432669393871*88970230145880244982401312177602815659065896037765606484677123557913151049*6906100861906711042294651808315507604303950918775090357302270975396916428842528615127185377473*108812319169865611113499765242170244114701137148935942205033082816725856614419925622882473381377372631922944203113004898243458798930001960412143970055880623982022558497599
     n+8 = 2^3*5*109*106877*271489*548347*1889029*690425747753501*7128686386107336359*61723043861290773050031488327281*327431229614131117772785108403020409*42690181228630617580877485609562883338265519449446054930706854035269082610743203*32324931823745846776555676474953076385481167075445075674853087885869075715823923787917*3364463967918569877412428970831620168291007106120572015124936370187375287372651445665862816100243713642209184736833502743289161172062635744244868288210173829656917423066932066331645933242315304125288886809659520629140738792514459
     n+9 = 3^4*31^2*271^2*1213^2*290717660692056178627380230059^2*55183042568818223466137174736479621212999091619327075187^2*3286538828791738865769644759971695914014392529618376267704168559^2*50874028544198389096448371570878800888606150471855795807934093984638655897441174063482462519436790916281^2
     n+10= 2*2099914577*969572514811180033060969*p488
     n+11= 131*307147*5268867920473195353617881*p489

     n-1 = 227*4373*1190082265745281*c500
     n+12= 2^2*3*61*631*1783*88903*46906094056543*45467031493033267549*c474
     n-1 and n+12 are unlikely to have other prime factors below 10^35.

     There is a checked verification for k=12 in
     http://physics.open.ac.uk/~dbroadhu/cert/ifac521g.zip

k=13: 500 digits.
    (n   = (2*x^3-3*x^2-5*x+2)^2*(4*x^6-12*x^5-11*x^4+38*x^3+13*x^2-20*x-6)^2/16-12, where x = 2^92+227683166)
     n   = 2^2*3*1022689*11589211679*p482
     n+1 = 5*5261*14307105795982965290513*p473
     n+2 = 2*13*227*1889*p493
     n+3 = 3^5*11*37*73*89*1697*22111*24379*50993*75403*294431*791899*7143163*37162403*40238771*318066373*774022187*1420502627*319986975727*360802459587989*4907888847800929*3948020931923766526433793457*1069158660766613132724682792996050143417*1026367444671582478750340776100892291660437*269849138487532374198877021349930016361247184220695675812643691485179934127024489200988726183*9828027147972763940840419206437324649030146749137361238797661494445753505040629992175660612970979552501716524157244914789537218745479849573839100489653483518323130703
     n+4 = 2^3*7*103*167*1823*9436961405541853927*2442634599251757965491697450488141846145584325007794753532934341089*600199651586709792321245759194699649871225663594379892041934006939971309607*527494574388111009719149625221156839723560610238541931586701578366691138991354382424573311558118964044710317560296491439030482498012276207040809762647265709450370220697759850439666381328859614847517666066051906937892145477551456225050086697670164991720248902858139512339587289553072087611949432974971329045183361427759949669197151
     n+5 = p500
     n+6 = 2*3*5^2*29*2664932713*64053375527497*12950177265791031072639356885920439*34173198741742332396716976271692371701489770814403893768924387440612462806903842625552319985422420268803792535665370032811*38999042013591597497050209246905972783624407330496158969655866746662086214865227946049360131542356116951118562262174103446007490605710219381435049081471459942844240208575133436379007979279145337209263917682421878854593870018670730188222715372117686044826074644600648361727317428823657930132716062226843897099648501007
     n+7 = 4001*33739*p491
     n+8 = 2^2*829*26342725653094583*1470343579963822912245711071*187007870852441168222504281550299*99610574090928502236269502589968168206515652266513541*113895390854000729877269012409350207774928221694641996544761053*4913497600365917962155071410041034922653694031638944416398503899346474340572860479110111*9571423313604982533704828521048066928977532586908807859259654574404430464964151003988095630328625510537874340855221392615775029906318151824100974216194570356545874213060957042918269033449215300619413721816385325826807
     n+9 = 3*61*1019*14243*52883*269749*73255439*1462827239*7462473053081018587763*1071679548735911780408902157698216943*36445304291066947549904502068646535721336497*484092028964579690545000872590192774860759892031294832503076409560989933338876408875527131*50995613413068475078730791162445890545117220641170507071916755522357759495828448552610940703376080579233279139258127608797223*438679328453087683131899883194844964328056296306402874587431956843033351755942263661835474320624975100141975738613328658670284212164843893047348661
     n+10= 2*31*71*479*601*2521*10079*11351*5160785634557467417755921932864257*12956653104462928747086439183606463*2976475056377687756545777040724081181425791*1111195533664096264407102983373638795453449159*1411236536953344109648201030313903202292715855689198457423066135444942202534466870230539797279*1202453802380202612900570567249140337265977386730165272588662040901323849540240433058583909567403721087774087063*93428370649115500684161484486657066107657288857729057476468058018820851220113189554394219484855663733342770455665657679
     n+11= 5*7^2*19*2039*2203*18917*20773*128663*206197*337283*4205437*4253763823*10361106142511*48219445919633*38944767299226517*823146395611037243*2567947475640367903*1980406199705343784496335146833*4068205927282173242680543824629898847404490879*170724646709285476974463174780574714342378046069994801064788497187739411448501668215598461179709025266247*172234777294748971503064594510341553906830528975589697516125816488729556569330134189448072867155899116079219277251511588963969376789767369662120332326617391898100471591557566930797914101561
     n+12= 2^6*3^2*41^2*59^4*199^2*3907^2*1067282959^2*90695686907824373614544935809648303855581556170634564267689^2*1045590928167366905609128138910595915667531392289262793343319247317^2*13281367071696254893447093955890080700243546027305640807619578081114123230285603623403537002718792146591^2

     n-1 = 7909460491591*854342030873171*c472
     n+13= 17*c498
     1700 ECM curves with B1=1000000 and ~500 curves with B1=3000000

     There is a checked verification for k=13 in 
     http://immortaltheory.com/cnt/verify_k13.gp.

Factorizations for former records

k=2: 23249425 digits.
n = 2^77232917-1 (Mersenne prime)
n+1 = 2^77232917

k=2: 22338618 digits.
     n   = 2^74207281-1 (Mersenne prime)
     n+1 = 2^74207281

k=2: 17425170 digits.
     n   = 2^57885161-1 (Mersenne prime)
     n+1 = 2^57885161

k=2: 12978189 digits.
     n   = 2^43112609-1 (Mersenne prime)
     n+1 = 2^43112609

k=2: 9808358 digits.
     n   = 2^32582657-1 (Mersenne prime)
     n+1 = 2^32582657

k=3: 200701 digits.
    (n   = 18543637900515*2^666668-2)
     n   = 2*(18543637900515*2^666667-1) (2*prime)
     n+1 = 18543637900515*2^666668-1 (prime)
     n+2 = 2^666668*3*5*43*347*16785299

     n-1 = 3^3*89*c200697
     n+3 = 17*111767*121224109*c200686
     
k=3: 200700 digits.
     n   = 3756801695685*2^666669-1 (twin prime)
     n+1 = 2^666669*3*5*43*347*16785299
     n+2 = 3756801695685*2^666669+1 (twin prime)

     n-1 = 2*c200700
     n+3 = 2*131*c200698
     
k=3: 100355 digits.
     n   = 65516468355*2^333333-1 (twin prime)
     n+1 = 2^333333*3^3*5*13*37331321
     n+2 = 65516468355*2^333333+1 (twin prime)

     n-1 = 2*7*43*347*91579963*c100341
     n+3 = 2*11*163*63530743*c100343
     
k=3 with a prp factor: 100355 digits.
    (n   = 37581121569*2^333334-2)
     n   = 2*(37581121569*2^333333-1) (2*prime)
     n+1 = 5*13*73*5743*2342706941*prp100338
     n+2 = 2^333334*3*36319*344917

     n-1 = 3^3*7*13554581783*14805287479*67956715507*2168351203043*c100309
     n+3 = 11*257*c100351

     The k=3 record for proven factors at the time was a slightly smaller twin prime which also had 100355 digits.
     The above prp record used a prime found during the twin prime search: 
     37581121569*2^333333-1, found by Mark Simpson, NewPGen, PrimeGrid, TPS, LLR.

k=3 with proven prime factors: 58711 digits.
     n   = 2003663613*2^195000-1 (twin prime)
     n+1 = 2^195000*3*7*487*195919
     n+2 = 2003663613*2^195000+1 (twin prime)

     n-1 = 2*23*173*3863*1954173900202379*3612632846010637*c58672
     n+3 = 2*5*35289796219*c58699
     GMP-ECM with B1=3000.
     
k=3 with a prp factor: 74595 digits.
     n   = 777*2^247788-1 (Riesel prime)
     n+1 = 2^247788*3*7*37
     n+2 = 11*754121*prp74588

     n-1 = 2*5*298897*17787571*c74581
     n+3 = 2*19^2*c74592

k=3 with a prp factor: 74288 digits.
     n   = 1363*2^246767-1 (Riesel prime)
     n+1 = 2^246767*29*47
     n+2 = 3*5*7*13^2*743*prp74281

     n-1 = 2*3^2*6930977*131315467*70439012053*c74261
     n+3 = 2*834797*174218530427*c74270

k=3 with a prp factor: 67218 digits.
     n   = 2347*2^223281-1 (Riesel prime)
     n+1 = 2^223281*2347
     n+2 = 3*5*197*1213*1783*prp67208

     n-1 = 2*3^3*7*17*431*2357*45611381507*c67198
     n+3 = 2*139*4153*29183771*c67205

k=3 with a prp factor: 64868 digits.
    (n   = 3045*2^215472-2)
     n   = 2*17*1151*27067*173473*prp64853
     n+1 = 3045*2^215472-1 (Riesel prime)
     n+2 = 2^215472*3*5*7*29

     n-1 = 3^2*11*13*22091*417293*c64854
     n+3 = c64868

k=3 with a prp factor: 60222 digits.
    (n = 38602791*2^200026-2)
     n = 2*(38602791*2^200025-1)
     n+1 = 577849*2645749*3427009*prp60203
     n+2 = 2^200026*3^3*1429733

     n-1 = 3*101*339541471*c60211
     n+3 = 5^2*7*397*49523*c60212

k=4: 10673 digits.
    (n   = 245363571*2^35426-3)
     n   = 3*349*111602773267*953666301013*43440278284896679*p10630
     n+1 = 2*(245363571*2^35425-1) (2*prime)
     n+2 = 245363571*2^35426-1 (prime)
     n+3 = 2^35426*3^2*1117*24407

     n-1 = 2^2*5*720418081*209358848887*c10652
     n+4 = 5*7*17*67*137*c10666
     74 ECM curves at B1=11000

     Primo certificate for p10630 is in http://www.ellipsa.eu/public/primo/files/ecpp10630.zip.
     (This was originally a record for k=4 with a prp factor allowed)

k=4: 8193 digits.
    (n   = 378149751*2^27186-2)
     n   = 2*(378149751*2^27185-1) (2*prime)
     n+1 = 378149751*2^27186-1 (prime)
     n+2 = 2^27186*3^2*7*17*353081
     n+3 = 5*1019*24923*43651*989239*p8174

     n-1 = 3*53*127*7043*626921*24117041920337*c8166
     n+4 = 2*11*561139037952529*c8177
     74 ECM curves at B1=11000.

     Primo certificate for p8174 is in http://www.ellipsa.eu/public/primo/files/ecpp8174.zip.

k=4 with a prp factor: 10043 digits.
    (n   = 21996007*2^33337)
     n   = 2^33337*11*29*53*1301
     n+1 = 3*5*1129*prp10039
     n+2 = 2*(21996007*2^33336+1)
     n+3 = 21996007*2^33337+3 (prime)

     (n+2)/2 = 21996007*2^33336+1 is one of 28000+ primes found by Ken Davis
     during an AP4 search (4 primes in arithmetic progression). It is a 
     Sophie Germain prime since n+3 was later discovered to be prime.

     n-1 = 13*191*1879*13153687757011*c10023
     n+4 = 2^2*3^4*7*258653726077*c10029
     74 ECM curves with B1=11000.
     
k=4 with proven prime factors: 5638 digits.
    (n   = 25390425*2^18703-1)
     n   = 7*15569*150611*1338793*p5622
     n+1 = 2^18703*3*5^2*43*7873
     n+2 = 25390425*2^18703+1 (prime)
     n+3 = 2*(25390425*2^18702+1) (2*prime)

     n-1 = 2*4657349734574297*c5622
     n+4 = 3^2*13*47*10528525069*3236067162807210923*c5606
     221 ECM curves at B1=50000

     Primo certificate for p5622 is in http://xenon.stanford.edu/~tjw/pp/p5622.zip.

k=4 with proven prime factors: 5257 digits.
    (n   = 297079965*2^17434-1)
     n   = 4363*22567639*p5246
     n+1 = 2^17434*3^2*5*7*13*72547
     n+2 = 297079965*2^17434+1 (prime)
     n+3 = 2*(297079965*2^17433+1)

     n-1 = 2*4846109*10719641497868929976071*c5228
     n+4 = 3*11*c5256
     221 ECM curves with B1=50000.     

     Primo certificate for p5246 is in http://xenon.stanford.edu/~tjw/pp/p5246.zip

k=4 with proven prime factors: 3500 digits.
     n   = 136857*2^11608-1 (twin prime)
     n+1 = 2^11608*3*7^4*19
     n+2 = 136857*2^11608+1 (twin prime)
     n+3 = 2*31*24337*674501*23226188779*3840896415363899*p3462

     n-1 = 2*5*67*199*14683*171937*541573967479*c3474
     n+4 = 3*5*338817126533946144966094117*c3472
     220 ECM curves with B1=50000 and 140 curves with B1=250000.

     Primo certificate for p3462 is in http://donovanjohnson.com/primo_cert/lcf_jka/p3462.zip

k=4 with proven prime factors: 4187 digits.
    (n   = 240819405*2^13879)
     n   = 2^13879*3*5*3209*5003
     n+1 = 240819405*2^13879+1 (prime)
     n+2 = 2*(240819405*2^13878+1)
     n+3 = 3*13*43*358877*p4178
     ((n+2)/2, n+1) is a CC2 (2nd kind); a length 2 Cunningham chain of the 2nd kind.

     n-1 = 11*19*11405214139639*c4171
     n+4 = 2^2*1123*9461*133900079*4881536249857*37975791341*3460283980865333080373*c4126
     221 ECM curves with B1=50000.

     Primo certificate for p4178 is in http://www.geocities.com/scooters_primes/4178.zip

k=4 with a prp factor: 6223 digits.
     n   = 2989530439*14489#/5-1 (twin prime)
     n+1 = 7*1123*380299*14489#/5, where 14489# = 2*3*5*7*...*14489 (a primorial)
     n+2 = 2989530439*14489#/5+1 (twin prime)
     n+3 = 2^3*943127*5020192965913*prp6203

     n-1 = 2^2*5^2*47303*76425929*5735860488526391*c6193
     n+4 = 3^3*5*c6221
     c6221: 20 ECM curves with B1=11000 and 40 with B1=50000.
     c6193: 19 ECM curves with B1=11000 and 42 with B1=50000.
     
k=4 with a prp factor: 4187 digits.
    (n   = 240819405*2^13879)
     n   = 2^13879*3*5*3209*5003
     n+1 = 240819405*2^13879+1 (prime)
     n+2 = 2*(240819405*2^13878+1)
     n+3 = 3*13*43*358877*prp4178
     ((n+2)/2, n+1) is a CC2 (2nd kind); a length 2 Cunningham chain of the 2nd kind.

     n-1 = 11*19*11405214139639*c4171
     n+4 = 2^2*1123*9461*133900079*4881536249857*37975791341*3460283980865333080373*c4126
     221 ECM curves with B1=50000.

     prp4178 was later proved prime by Matthew Peets with Primo at a time where it gave
     the record for k=4 with proven prime factors but no longer for prp factors allowed.

k=4: 2135 digits. Same as n+1 to n+4 in former record for k=5.

k=5: 2158 digits.
    (n   = 7096755082*5021#-2)
     n   = 2*(3548377541*5021#-1)  (2*twin prime)
     n+1 = 7096755082*5021#-1 (twin prime)
     n+2 = 2*5059*701399*5021#
     n+3 = 7096755082*5021#+1 (twin prime)
     n+4 = 2*(3548377541*5021#+1) (2*twin prime)

k=5: 2158 digits.
    (n   = 4790484140*5021#-2)
     n   = 2*(2395242070*5021#-1)  (2*twin prime)
     n+1 = 4790484140*5021#-1 (twin prime)
     n+2 = 2^2*5*13^2*1417303*5021#
     n+3 = 4790484140*5021#+1 (twin prime)
     n+4 = 2*(2395242070*5021#+1) (2*twin prime)

k=5: 2155 digits.
    (n   = 14635080068*5011#-2)
     n   = 2*(7317540034*5011#-1)  (2*twin prime)
     n+1 = 14635080068*5011#-1 (twin prime)
     n+2 = 2^2*7*19^2*103*14057*5011#
     n+3 = 14635080068*5011#+1 (twin prime)
     n+4 = 2*(7317540034*5011#+1) (2*twin prime)

k=5: 2139 digits. Same as n+1 to n+5 in record for k=6.

k=5: 2135 digits.
    (n   = 447295839*2^7061-3)
     n   = 3^2*5^2*90439267309*152397278879*p2110
     n+1 = 2*7*11*17*23*31*43*24633501044437*5238873869796767202503*p2092
     n+2 = 447295839*2^7061-1 (twin prime)
     n+3 = 2^7061*3*347*429679
     n+4 = 447295839*2^7061+1 (twin prime)

     n-1 = 2^2*271*4273*208933*567367*25463453019262592726380909*c2092
     n+5 = 2*5*103*15241*c2128
     226 ECM curves with B1=50000 and 120 curves with B1=250000.

     Primo certificates for p2092 and p2110 are in http://donovanjohnson.com/primo_cert/lcf_jka/p2092_p2110.zip.

k=5: 2063 digits.
    (n   = 1749900015*2^6820-4)
     n   = 2^2*(1749900015*2^6818-1)
     n+1 = 3*173*2707*7207*p2053
     n+2 = 2*(1749900015*2^6819-1)
     n+3 = 1749900015*2^6820-1 (prime)
     n+4 = 2^6820*3^2*5*17*37*211*293
     (n/4, (n+2)/2, n+3) is a CC3 (1st kind); a length 3 Cunningham chain of the 1st kind.

     n-1 = 5*13*759476989741*2089473326209*5861811536579*c2024
     n+5 = 7*19*51610785291067*c2047
     253 ECM curves with B1=250000.

k=6: 2139 digits.
    (n   = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-3,
           where b=5026578700, m=13416739015680*b, x=(2^67+28683395)*b, y=(8*x^6+24*x^5+50*x^4+54*x^3+41*x^2+12*x-148)^2)
     n   = 3*5*115523*2364592753*16987001959*6805713408228112745527499*p2088
     n+1 = 2^2*7^2*462899*p2131
     n+2 = 191*229*277*317*353*421*509*919*2243*5879*12377*36901*76607*80471*224629*266059*329941*380951*904919*2136311*3056057*5914483*8096183*9465601*21550513*94799377*130860217*155882081*45822412859*8146655372227*1474846703267549*14278046242057967*30669562031647339*50840461718040961*92756515760205629*52290238661226907927*96677437656827057417*1024366806097559014891*11420007000929879319421*56573337948259077277921*1112138785081162287658151*8528010785159544755667973*36811892890610855550478433*1097776113917720517099045197*6266573674771665058327422803*30451289066171237974920654932085755371679*1316506222346661036532477000837519412493197*111403796509772263652769724399362571727608328017*3624284692109954237076203309579972669010531411749*231489152760278164551457534708931642648344475815232745703*147116374129195711393035548788265178287722577822324906900753*20859830985834684423322955342287747410510371603352932446294829*7904406721787550323058091000193526797479128058982617278101662873*46540398168652715463731610192037761520603950804914554357304827651*257704475111226397986479160737832082626014496927642276922664061017*1633827832645400817312925190638479419466129452872018361627854113293*382662618255823283472179277788367200218513195283130424126700215295748735665441*1391507993148700240606140015038276134257033549860873095114389589694682942562789*7499469611937904980182356228537145717109317190667055527703147049893120442453955199*356484869506390402253567489057105023111650744085980620919627955814007320376395156783629*4914377652267665655490741867052323842264433563348177699667952493668432484240598433717557*378510207963271339787785564166177590272761189694519309918631082645427918103057025277821365211654046232903*10882730321922719433671807500165889524951973748667067180984450031864886847971033873122407056582166336612089*42492274401367782319359911698875554225735898020880648763949946800237401567377284028369105348871780497889130827*550436717882902423471815618746320309019776404049358761651392910282277600444836241713723132473766354880050594580270621227616367362518126662367*5114238400333688448457232468331562020643807865101179717783965219897043019763899891052712357151409461393105479520684052940760427498538444060680207904388052821822493913325171
     n+3 = 2*3^4*11^3*17^2*41*53*71*89^2*131*139*211*251*409*593*1277*4583*9769*18289*49019*54371*67079*167381*286009*463717*10388629*17815571*66515707*698686561*818125109*831161249*846265213*1212778709*2355934153*3058330189*4150299187*25306735499*3604486352897*14948583710353*23141076969449*153460724640233*334741271181311*512852123180983*1055641941839291*1392151174305467*8867626867246039*9667517060163091*182124553316524501*202781152357277099*406336869368999861*24740829299338719433*29322679844588024497*24294379584087164777131*9581091219467836447210057*11623108404437621322717169*680338395030015030231632893*20518929682734143207132735353973*66563637308857605625389901982459*232385937495200968879386754974313*535735630734461728406216081733553247*508439874253891778606279503653232082681*568792881367546646342582771622531398201*8858542841440374529189506450521929269739*10843913415694740420736960889105018295370049040103*256316245276700640596341494983955113716708347284713*269332371965553888044556108484090910220885368743111*33712744122427700771320939883696582781318705265620304873*485185740258747937662437489843040968838887833679684367157*788833359586429565859990494779597014569532914058188795561821*7291571476603897113533712637384336734903709885014047061381079*19669397718658187975325767914206200463853657104127548035301433*3125382489664258590391298530427593160788815853502138689793042677*4328160415203489875259678235637742289270192182210119710370643850927*309822335403073024796762355934933497647408610442517613113885861757379*1103898050573311594319428118682109119879937737776005655410652663836349203752737*92486996234402935152972092853138208762410572617958701845300321611350478666752372301*500009341040688147102544811636097201842624578131324143223416593271893196262498696726469*63547811132003317055239013988352688043328652147686430763336739113148789999768204463467171793630031029*64319007646797325362848402230487990288109209313701267028024171110636741054514980740206560574125978219756541130779571*842856784319760594663011094715040960562959489469298103728654570968227035118953458504428238545469047549975587147482030615502207*37629800802773044149485257881001665650425553616023615308422357601471426709159760613987492291949078144731754876614032040655663376557577609451
     n+4 = 13*652739*p2132
     n+5 = 2^5*5^2*37*241*28843*794200385809*p2116

     n-1 = 2*6911*18959*675413*81134281*c2117
     n+6 = 3*167858252466989944843*c2118
     n-1 and n+6 are unlikely to have other prime factors below 10^30.

     There is a checked verification for k=6 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifacpte6.zip.

k=6: 1803 digits.
    (n   = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-4,
           where b=4011209802600, m=16812956160*b, x=5000001251617*b and y=(2*x^6+3*x^3-148)^2)
     n   = 2^2*71*173*1567*424064851577*4764539601163723243*p1765
     n+1 = 3*p1803
     n+2 = 2*701*19079*282279559*p1787
     n+3 = 37*43^2*59*61*83*89^2*193*227*353*563*673*773*1481*1553*27617*37633*58147*64951*205201*402769*435569*6568097*8204233*62792377*257777969*775643203*1687171501*7901714521*8611020113*44900871599*134596618103*194727326069*4054656560329*35310788241601*1144536016916393*547418495121864799*960543818610206749*7512682467497170148153*77122988632197567410243*1041998597004029214079279*3934208076568895847863441*76355031909623483424534049809013*9541464799334181477034188730576609*40869923748552947588194499753142151*173375138423053317576233169105066161*24960491263045151537938246748797732033*4046241609244762139507273437307875056559*934101794602481559648481462842733450340177*100820663465058033355399489727414672691672458489*10355778250337305458080594000027129114245922478341*157937875196459568563896915849429921595309413943803*35332528502521473760795720289180986110801399084551341*732696498696012722939913820432299305647881186929419241*208640843623123680312945317256315243660831682858757623357*76201886634791577171993998077475484980646225736074267481987*6438866769945748412687984544492447585022109014683560792252689*34210709690822198267146724151373988688494165650865775297103199*2132791641418224620482034613867224765473751663257207506429723889*4874057484841525794516697638376051106947984422695239381249223971*13560606627692164809189162883273527153385704647924894629000192913*3352506997242646456006759979730890421610858650126457579100616082691595385831349*102692044298884915205557323891340757406411668602429945398737447511025039561640393216003519727251263307*189922589414865226468622828424540278517145146779251204153423511423539426305694130190815480795318828893530509*21266168480080580156785583576426010575615342816000905263892120638719521665707623169105733601117777886837039611*55149006508549405130037208376035974596763660808047405369568008279354529871030586217152406508249456842538207077307963056464229847
     n+4 = 2^4*3^4*5^4*7^3*11^3*13^2*17^2*19^2*23^2*29^2*31^2*47*97*349*677^3*1601*1741*2917*3331*4229*9661*10193*13789*30853*38713*62473*150287*186239*646981*866293*917101*1698043*3746179*11497231*24666283*36191671*49429327*50339537*85334927*124264571*860249329*1288198913*7385526221*7385526221*7385526221*8477874479*1141359939089*31077751869103*326416583825723*523883179531643*3571319405698511*24185665817420131*33548677916339821*225006287244512797*395997081191384419*887523182980651073*1139672144933035073*7667770680246290221*23897473911488134597*1100989909387778738903*2625800907208358841833*131639598063226658510639*715781480481372739220274931*1180722659058952538240342286749*10685698868228397151327332353471*14460423523793974387623255510263*411857497472623648085321963833273*10876101946069142809741344894091519*16981907479154384669308748833397867*22203951185157442583966047769588057381*5402898330595406519134147306421071728281*231491914441581992784465543486871431668863*1025339487917652994806865866045830130341031299*31115421453323513037476889813038334113201214677*67804647002031445271110159102947426615620185783*12603537558996293938258626344198230911354106273722673*553954117005392087403276328493242186322972372392295239*4884969711235714491445156179048143737630467138460814567*2359583141365393112820375931702887501111092275711062284863*21683720997503404786034957307842654904006922132535428397388587*1194426526170945773717097618626048988920226654130164934661119087*3179321305537807099689388278906670933944837646065351852662778017*2970796650255750198546518165137105392831427224063820842847779446318121509591*40550699121998067419991842732804823585659593996219670981404869979254505478981432765437046291*68698278844643201913685198175034756270603298277530436699827937740857333506176487316952783366947983291613913*64775450186718835923061966374498704041978953377071501172637808318170164907715379477552698465322868941229874726228423297084374755363
     n+5 = 419*3407*p1797
     n was constructed such that n+3 and n+4 are each the product of known divisors with
     less than 154 digits, where the large divisors are suited for SNFS (special number
     field sieve). Such divisors of that size are possible to factor one at a time.

     n-1 = 5*239283371994839*543315317127619*4154925272246149*c1758
     n+6 = 2*8081*33697745047039*268690037677741*c1771
     400 ECM curves at B1=250000.

     There is a checked verification for k=6 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifac6nfs.zip.

k=6: Second record with 1404 digits. Same as n+1 to n+6 in current record for k=7.

k=6: First record with 1404 digits.
    (n   = (y-22^2)*(y-61^2)*(y-86^2)*(y-127^2)*(y-140^2)*(y-151^2)/m-2,
           where m=67440294559676054016000 and y=(m*(10^96+9581328)+22)^2)
     n   = 2*p1403a
     n+1 = p1404
     n+2 = 2^16*3^10*5*7*11*13*19*37*61*191*227*373*631*1009*1289*1811*12251*18223*19949*39761*40099*124139*156967*452077*2130173*2705537*3462937*7633753*8515597*33173939*52641283*56707439*2678042047*6028890499*235057646869*2293399729639*4716162386153*20290714976827*76966210128923*82930514647957*3414960948392327*4798235010101597*492989197272845783*1021070851282029949*346642803549673938493*10381965549080973912121*10705853500705134572111*3402602170075407160915063*3896113165479453193941681514329473*572982916505698266772046961556571719*7779627119389705894268583790178040713*2679272214955462246938889432925080340640361*49177705726337585346837755875445232529608396349*241826146812234755480438342468089149152034499363714121208377*243192829165886509367881231894318437733436233104128418423890411*28990423033538119642473011651390774174058997738460601175798278583*437141645229051532697099686819531598452795793496929224693672331927*1418753969662679659418666553265322031463146093384848016223510836379*645413449365286314824919986919822970362519337008974665550190920457255518759*20965538503194242905032903697401432404999762007708204375118749419811954970647961*3294297702148718077561555791777335845260219149060777243934402094257398262227750093*2406629749273883289769692163217837087029221186833486436482263322161068612926191024481*234004321348347878243450131076080120876538597870136557022678611648262729687220946497966699*2342885206975304600030966940136109235370201749576148604770764630664574779225197114034373688176651
     n+3 = 89*137*149*257*389*541*853*2591*2851*3331*7559*75329*93229*102667*225829*373631*1307347*7780769*13541653*99379921*188069213*226248731*1457666107*7019523803*69270603191*130601553923*471637135901*499457115919*6996512652487*90468395140231*126512656915647001*561436143437641939*39622341152320502977*5060294951719365191399*15507396773627356960691941*34802122844262613744836487*73113052821679897631882999329379*13881170525247452115444724778498657*11236972040374240265121033071750573780554937177928509*26291144543042185831481954556908348064015228448587607*5441441819431723433035024498287323261369539563023632649*3323253180187898782557645135868186230115978698950734843531*670187939956550896144016063406622725370069384221626353600796980706687*388508861296593678793647742126412846197330928103032998798526196672169751*46526361176976331131692448831886295654180937200509756317956476079377060276107798251*123723846994132061509262488616575553842266557999262274735157745365879370310957511849587*52777309311666259876826027372783385850666230122869905057157388781104477582284263792304493*214203286249223750965285191327202298873780724143364323737823298751073982098758638527694627227*30091809213543725941251060360120799012501436121378265374301342522682366334233176762803248560731781*1444362720210768757829252247484379543219331174080222949691438360399704398339766729979542816990814772477899*560896070384991947831892486100383451139489111138563275709977404579309957444584330524226413933472530198395863
     n+4 = 2*23*24943*13071241*p1390
     n+5 = 3*p1403b
     p1403a = n/2 and p1403b = (n+5)/3 are different 1403-digit primes.
     n was constructed such that n+2 and n+3 are each the product of known divisors with
     less than 120 digits. Divisors of that size are possible to factor one at a time.

     n-1 = 3*59*104882036089*123707274359671*c1376
     n+6 = 2^2*c1403
     320 ECM curves with B1=250000.

     There is a checked verification for k=6 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifac6a.zip.

k=6: 1104 digits.
    (n   = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-5, 
           where m=67440294559676054016000 and y=(m*(10^71+145589)+22)^2)
     n   = 2^2*11903*8908713059*20605508509*171399027325457*p1064
     n+1 = 3*1867*273269864239*3658698250513*p1076
     n+2 = 2*p1103
     n+3 = 181*p1101
     n+4 = 2^12*3^11*5*7*11*13*41*43*47*61*73*101*109*167*179*563*769*947*1669*4483*8423*8689*86857*249421*352441*1452419*4213189*5337557*38376203*75807749*1342644727*1592856247*79036647043*2066440556016253*153467969439714187*1015711703948543083*351255950120842197943*1181227353539948611763*1748360760308591989432139537*24015455632265221840820445967121*80309621927907102740140515572868133*109269974651161535058599454239640944990321*9517480444270086129756297023952251070704479*11624759710856712926151094525593318553483268019727*1624383839750483864275832116445201199733452749965743041098763*3321388188935613335625081797063937564073265903733973310154511*3930918517802853429791414852699425685138947223854804684150991661917*15700457668245089126937307919442053598829465908914062668377242055044331*613509075233122056292153808622241583039288732180711585974780991303046186587*2259724066520182819518427246354597484635432626048626394948984886210453682509*20145661260561675391127046332451100091532791360010608280969949945015502216917*284317035570965203892357347738074193719696256560671468387518710886460130970549678393*750409068179835827663191219069403815613515074981151907320017458837177845836411021913
     n+5 = 23*31*79*97*113*641*1811*2357*2539*3253*11383*11617*12413*460111*545641*22231589*30740459*158884357*346304197*9596667107*11982265853*172611395597*42341290844569*128209993016077111*233324021865808229*1067315559086224029308057*7756943116200250207544789*324725593355593676830982063*1518260580302799430092803189*14437709862395025185324753357*741946562546897516077231120190975401*1215947305584459125543131239649844633*2459003388630363360188816574544627010587*526712065096184394205028755707683695993381*2482021200509575214080292599399563875733646553*176252349918469581244355387429024446510384958783767189*829503790824119658573264002082945813894732504167507643704839870307*5114226172787067738433625217783426484550351898768325916811724886909*148389983113982515325263241221475679888839593884951677946444415461929182414531*55666846987356792972772284987772094350949848491397034769842185515267556045842978847*758457842480388689538457718990000165699372176981969017542726678962273839761971061113*58138184965237977600000000000000000000000000000000000000000000000084642802109040319208064001*518771496612892723200000000000000000000000000000000000000000000000755274234203744386779647999
     n was constructed such that n+4 and n+5 are each the product of known divisors with
     at most 94 digits. Divisors of that size are relatively easy to factor one at a time.

     n-1 = 5*1948359547*c1094
     n+6 = 2*11969*4471247659*460041067299095322777708682147*c1060
     700 ECM curves with B1=1000000.
     
     There is a checked verification for k=6 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifac6.zip.

k=6: 1037 digits. Same as n+1 to n+6 for former record for k=7.

k=7: 1404 digits.
    (n   = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-4,
           where m=67440294559676054016000 and y=(m*(10^96+10624986)+22)^2)
     n   = 3*820681*58279268867*8089381088597*p1374
     n+1 = 2*29*3317521*15351437*13973807284874590379*p1369
     n+2 = 1427*p1400
     n+3 = 2^13*3^10*5*7*11*13*23*41*43*53*61*151*191*197*277*617*761*1063*2251*5419*7417*34649*129887*148691*215297*249377*292601*310117*333539*392647*2602291*5158817*9098161*9572413*46347793*47181619*49687879*91731397*116976971*444476651*16651689479*141790225477*11862804584870311*475827842069624329*1952556888901014284396298871*1656027221545269854046521732323*49640146225618621543893578601492354823739*8682665993637814126334655623002945126132381*788345137743082394797513912882652922339198530323534223*46431605934975407105850843188010367507618025283882564529*3366617260222806408173455918182918611428964447899257888247375663*2719605820033375780663668494243928674314151927435135054868093296655402493*69099219944925942336457370755387365575325867696295211021371495432795990129*7423954758405991178947245926584616929504622702291806044646152378579768926683*507124864910740617317834708384624322829685165741783347184747283863236535136436949525978801651*363525565401269080033916712603583884338729712584004041380896373487530555465536191928885837667299146959*852341347139314030568300485338245515621488696837146351125562512551910877500537794424305520419431727471996047*3558804019347548057129979882963459534270211185413580250765768246811588239876718982576493662356513224206328931*230485523151582031558333703575859275942324188912546436956811494150737693992843496775472403431813922482952001036373*54651778411406850904376012965964343598055105348460291734197730956239870340356564019448947096071463154322721600683773
     n+4 = 17*67*101*103*107*109*263*373*463*751*6899*14629*22129*44491*234917*1146833*1859983*12292663*3489932473*4653304463*13228645723*56901587167*443097598171*2569113463417*6080736107339*205932998345467*240850667750249*509184985561331*6453823454403719*1545871005161244499*1775442683235787970389*2451694912860432213727*89792269186185778201159*1433951135487402561509092703*1476894477771977030320654201*7585718198499770808806342078801*281519710728123512693670143339827*2356521307346163098298477871916817733*52779008407884145440776317064241963137*6848307225277797080796031980759801261721*36978195796816375632920076819259670928713*5989972262101629030141145495263426612254919601*32113835899565282657830284793564132341737179238863*91609376828281820114635949438428467784649774584286839*36786583168467743260795894602757035563550833953360713366253*2075649148044335815479282359000619116345060219424797088278928237306672365791579*14817649457223344618882571026934739805193947170804011001801725368635753671216882070398683*661791067702497042286854813438731762743946616995889032833714288555760467878168944873689801*12113030945818324826368053939966187535827171168662412479040300135359529000321647451437672811*174608865305212009166718383113652644062482549528644096958004773623125241833539831576705915580917*486425920865949537489015750397803498829412176281597674228671188335610801241814848467600911390599*2823160223584328544758003137642306989810953614879045695865510201460912967585227792523246402587435978803
     n+5 = 2*p1403
     n+6 = 3*26535329*37907509*1185659957*45446838221*334012213438151*21633422296379581*272780466084810349*p1320
     n was constructed such that n+3 and n+4 are each the product of known divisors with
     less than 120 digits. Divisors of that size are possible to factor one at a time.

     n-1 = 2^2*37*113*988979*2759209353591763*2338830543576644077*c1360
     n+7 = 2^2*480023*1408111*32195489*5342864083947636432359*c1362
     400 ECM curves with B1=250000.

     There is a checked verification for k=7 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifac7.zip.

k=7: 1037 digits.
    (n   = 21247003564*2411#-1)
     n   = 12906420959*p1027
     n+1 = 2^2*103*51570397*2411#, where 2411# = 2*3*5*7*...*2411 (a primorial)
     n+2 = 2524541*p1031
     n+3 = 2*5002841*6245491249*p1020
     n+4 = 3*485475518243*p1025
     n+5 = 2^2*(5311750891*2411#+1)
     n+6 = 5*3691*23063^2*2961991*19076087*3778442561*p1001

     (n+5)/4 = 5311750891*2411#+1 is one of millions of probable primes found by Markus 
     Frind and Paul Underwood during an AP8 search (8 primes in arithmetic progression).

     n-1 = 2*63501029*1676569836074094735760183*c1005
     n+7 = 2*3^3*44041*50036011*c1023
     453 ECM curves with B1=250000.

k=8: 804 digits. Same as n+1 to n+8 in record for k=9.

k=8: 703 digits.
    (n   = (y-23^2)*(y-24^2)/55440-6, where x=(887040*(90^12+111833012))^3 and y=(x*(5*x+9)/2-31)^2)
     n   = 3*p702
     n+1 = 2*11839*46499*87049*7446793*12390250603*p672
     n+2 = 17*19*41^2*103*137*157*163*241*269*449*4079*4133*4721*4861*10753*2351263*2485073*3057007*57482213*68219878477*381488038681*462620692403*12683050903123*411332105162957519*3276055668231296267*6870702102249368659*9669550233544435375557769567153*2360578532615379194929178201025349*3892682480172514970480939619480019223*10494347741384920661144877561340083568978429272003460742633*1829055837237982915251619336597180975905571357734604112099369675323879*9867819978384278619956294537368978925192159544086646398961374957292063727*3935161077451634316971150930230328910277288574512329708155729392891380404108714093*1153403655532048033020388150472523156118899935315742577335109467337017712352307759295113803534362481448755802478888663421616880604044232388184863656791
     n+3 = 2^26*3^6*5^2*7^2*11^2*31*59*83*89*239*293*463*1237*7573^3*392443*906823*981742453*140477207564839*9323568482800743161^3*7795461012843187990059199*367976472250843280791966711358177953*2369898157201664369071987219354981398749094340205776257348069340155257087*8586003417616576718090324916324707595976594797359752098641043332748650167*8735493552188193449423855866811330500338440890721691851514216931005883399263423037440001*3170970794225727329321200288517180338107872852190460711796341512855639339227445852183298386837309339215974029658316649854721145406061361*23986911952889385545207942919750495805093443987293237095248265406287411769986943478571076469349420621340301274180158830039251291653843647
     n+4 = 53*173*p699
     n+5 = 2*257*16001*p696
     n+6 = 3*13*23*487*641*677*1319*6961*55609*70507*116293*274973*392159*14237603*68584667*303775457*425585317*13728113437*131679928339*9654557976929*12435367768589*630259423347317*901218628584317*164149758593811967*1988600182502075783*33149325702335458851579341*58130123165064994906452158476554824029*15223909874348358974851188325731198722002409*15108855722066292086389891853338191325033709928096121*499524512241167061789413229003729509184108096802211073409*42823638155592295501589956561528346972325869068174232243623637347*16951154355788530178104217461024513706300145304574366796538557804121*37680223587148534204317012443876372557929839640420906180155846961328886566664252556913281856156597565057740680394042206583621688028269799636710358518080817004999965407
     n+7 = 2^2*197*8429*313933*260172449926811929035350826091*1269872752096078507979335383599*p631

     n-1 = 2^2*199*66067*5519123*24972481*c681
     n+8 = 5*2993423*124255771489*142190675346823*c670
     700 ECM curves with B1=250000

     There is a checked verification for k=8 in 
     http://immortaltheory.com/cnt/VERIFY_703.gp.

k=8: 600 digits.
    (n   = (y-23^2)*(y-24^2)/55440-6, where x=(1320*(10^22+1932187))^3 and y=(x*(5*x+9)/2-31)^2)
     n   = 3*7*12203*17669*1099168540978153*p576
     n+1 = 2*778763*p594
     n+2 = 19*47*73*151*163*1039*15619*92233*4891127287*22577760473*47111633003*12070429199483*199006571381687*505906662595487*1268153950947427*3112861189877545933*5854318051903813309*8967446026650440921*412153717834508405396923*2152209086997284753405165939621*37831708440169494583827256188208582483*8249904673643870473934306022816076007428663*9739353386493274920005734938730781351211448586959781*2446244936377136793664896517135503616925585329882636307*275172535645132588474698760948412898025635380628964475577089*57790582804607151344983828096115383604868580390780704722813738821*1671556500708834606767822131230920740407412585120283129906232745727191
     n+3 = 2^5*3^3*5^2*11^2*13^3*17*23*31*67*1063^3*1055933*2582369*112864327*120841073*943018301*4214057027*26688059579*159090509027*267473033453*41857703804036059*723641363340328673*723641363340328673*723641363340328673*201583177283742768350363*6870502442454265018806723641*27596850040722511032814742539*355792854625524465962588947535325171255860024510489693551*896071701079444588332473182556959050185892140360842237573*484882078319894128785085629548448989511694520807645981942364330906558109195524376307681*6262060188743043039415010743222646616899150266572151851834807968161857158595536784536792596382842491016647411358484563906157076210321227747973493
     n+4 = 461*613*p595
     n+5 = 2*2621*p597
     n+6 = 3*107*757*4909*6229*16447*65381*808187*177513353*296694881438759*10622943310153609*144830170731359871913005718268119*13994754697828882745886899442981334542757579*261939634522000611741702503697485328141259777378173867459727*5749920000000003332976202512000643993328980305415877184611415643348193760007*30575759654977077510752353461475337956997642899877800633700100675711794525297187617*11413913219685676030478828514755884493375522838318406507453170587599985178733468847520565119741918049927971765982974521*143338764107655986839549495454178765826740836571371361424355522329730096011609356106622869993752761596283471508374706064109
     n+7 = 2^2*7*643*59407*23355487*p584
     n was constructed such that n+2, n+3 and n+6 are each the product of known divisors with
     at most 152 digits, where the large divisors are suited for SNFS (special number
     field sieve). Such divisors of that size are possible to factor one at a time.

     n-1 = 2^2*25711801*125018686680668134578227*c569
     n+8 = 5*281*c597
     1040 ECM curves at B1=1000000.

     There is a checked verification for k=8 in 
     http://physics.open.ac.uk/~dbroadhu/cert/ifac8nfs.zip.

k=8: Second 509-digit record. Same as n+1 to n+9 in former record for k=10.

k=8: First 509-digit record.
    (n   = y*(y-23)*(y-41)*(y-64)/55440, where y=(13860*(10^60+720251))^2)
     n   = 2^4*3^8*5*7*11*13^2*37*43*103*817793*173899823*22221761707*37836465211*1128969422473*7226426977433268109*666555585664696017970179047*666555585664696017970179047*4274216449873110505368747152083*4274216449873110505368747152083*924544586620285060733647007145063151*479490073146871050534575062193704807941737557*3752992276532013774050014526575133031229811053*7291083234967253095183997555458638002935444601743274397222687965174503227040871*228474332808418653912215340656711947252683332494996254331989692206925564018759152735292109921745527382957201
     n+1 = 19^2*79*223*17021*129341*187387*470077*1134503*27596381*591071681*219722506332743*823024421902436629*7217017062935103051815890914354049*61551757785499981066434091025664731*62152466367713004484304932735426008968609865470852017981985107*149767418497179338435882188243580843563124341953736118862837592294108775607496189*11456039860137004758584981278594652001673176571216011518385988939472768725293749566253855701651241969269*125637734984402123698410978366257187832009797939597866455282183288466661002745093507460307485903514650852713133199
     n+2 = 2*269*2647*5839*p499
     n+3 = 3*77863*p504
     n+4 = 2^2*349*727*1579*14407*140123*144569*419527*1125823*280638781*12482765623297556507*293244109006399769051*3924302134130214471389*37094900399021319425878133*8628414443325319794582933283*22394618831576964656662185799*85606406273330155008249155057*373302185646651554489812114237*1523737310190931395809246034113569*81026239243914061907529877840939545661*7541626939748321863588285645973660161581613688604461*43384942263778101127544397295737593104298414085317202239*2980895151661693476335327178562617958496303370683854299520305985770282863620140547091
     n+5 = 5^2*17*2895637453*21087042971*35055489406115209*p470
     n+6 = 2*3*2879*36931*p500
     n+7 = 7*1823*38167*331906427*168969463248297611*p474
     n was constructed such that n, n+1 and n+4 are each the product of known divisors with
     at most 128 digits. Divisors of that size are possible to factor one at a time.
     
     n-1 = 193*34849*44089*1520083*2478731*2289665947*c475
     n+8 = 2^3*902141*1832165329*21321528412173053*c476
     1750 ECM curves with B1=3000000, B2=4*10^9.
     
k=9: 641 digits.
    (n   = (y-3^4)*(y-2^10)/55440, where y=(x*(5*x+9)/2-31)^2) and x=(1320*(5*10^23+7574922))^3
     n   = 13*31*73*593*757*11471*12487*17041*54949*278029*901249*4790389*29063737*216219313*480617453051*730400840441*54101370630581*33174660281920153*1200313583779857538637*2021944779729779239081*2837368867353968687549*1061625261781019653637690586250321*19340092075750156710130380623411729*2061984639973675784172435043748808101*156392771603960271036296396091376763615503*2526909010888691858842287468626436550362106081567*3724660798182483488936637129340557412622657092030884777136438810213*71874000000000003266639662968000049489081298177777241917287123570268565196416001*126651843875397099513662125701570217778163140409122891242823544251509932501420396712305178111145609544574982476513345693771
     n+1 = 2^8*3^3*5^2*11^2*37*89*977*2393*132647*41511931*99852207041*587299388795151066599*250000000000000003787461^3*1778007979566865640487680539*90713012204379285603476979477401*590488790619001905875488546671218429197711*17805009757329579532871514385178076366829311791*98905616340494743760280350646391078198892638531*71882161957120343138113743329439944242294214201425931996660692604102233*31757568132349259959665435705861571851253708159452840258558330447880606873458258600929*105749680163766642161165164223625010949665222643768206062270100098131812633714796206947998898813574041697679463706122684435120868673908214824392451997940519697
     n+2 = 7621*10607*83911*p628
     n+3 = 2*p641
     n+4 = 3*19^2*23*43*67*419*433*557*5839*11005151*48791387*248743477*173166075757*176767927013058599371*3664337605725212245321703*38481802057600502530496251*263247353722581788826308137*10209898961843153162557563806797351795337*400438949478601771715861066925877874810624450405554075346315813*5481394579601493765715335676312985509068059008928260228225197198334163861140908690109*59671284938991446654929334084586000491614000771235438219780687791993969485740039897984101289374576781389292405323136206123396283987378378663819*3826571760000000347831791312832653173993441574924501382327329177624991886782939107192155367560772660694023438371884513433078262544237345403090052800073625471999
     n+5 = 2^2*17*281*p637
     n+6 = 5*7*71*3027473*p631
     n+7 = 2*3*773*1699*10103853977418233687*41681046033582287059951*36621959455077044549010329*4211861143455373066167093525382847*86528790939635630219352122503699213*p498
     n+8 = 5960137*45290243150024180117*p615

     n-1 = 2*7*88309036301*7938470054221217579*c610
     n+9 = 2^3*1138534864709243017*2353150702179440759951*28486552286765416493593*c578
     n-1 and n+9 are unlikely to have other prime factors below 10^30.

     There is a checked verification for k=9 in
     http://physics.open.ac.uk/~dbroadhu/cert/ifac6419.zip

k=9: 552 digits.
    (n   = (y-23^2)*(y-24^2)/55440-7, where y=(x*(5*x+9)/2-31)^2 and x=(1320*(10^20+13065906))^3)
     n   = 2^2*85580203*p544
     n+1 = 3*p552a
     n+2 = 2*7^2*16127*p546
     n+3 = 17*19*29*47*107*2381*4817*4877*7243*8123*15349*351859*64106027*617354687*58939383403*2270969174255932789*64599558784499658763*70581857953182048626663359*832941909454872901874063981*38463978481445185081160367469*18565176607267555386412648342877*72712545267645351975447576546899*1872363765601959731971246441364247991*2945265097696419515747753341191750275309*316490425386673017172951555041414472713325780025891*5291973122549848056044955804501530796430017642474721*2451968078534651120722918940489036434974865520343721698595446659*27854698184101546744112650299911007030936127637904783943133412329
     n+4 = 2^8*3^3*5^2*11^2*23^3*43^3*73^3*101*233*439*1579*1721*5641*5791^3*7001*25609^3*3059233*35685077*200804353^3*7372068659*5598060112993*23151817056913*38662729757040799*376176861511712141*4301348426994983857*15154667761484246406148307*1033141326087248524232877319*10617791649528225223660146631*391417105842348028109572403541777737773082843*4804384929120809533054017484874238511273037729623*1411936758329397253090018710058691271956742873966000085028776838783*831403915444995114442474420649091367670644872564910283524087531524227782803443372416623883166831227427855923684347833
     n+5 = 223*9887*252181*p540
     n+6 = 2*p552b
     n+7 = 3*13*179*4651*4691*5419*14503*20483*45007*157291*3287266267*65755593703*22579302731919529*22650433627111273*254320536721426607*12212080029376746054763*174550854747787142801760373*404207064226822367082013951*452888967550655599403298024877*2271339091394769438948112369926152025853*336542104860088953870195699074182766905880245025434197664052101397215693118076537643469*507854227981676607382498641611332244205705154434984017560541283876746490947606161704323477*74631722951875434426641848605731026702747965564034236317050130044910555530463123439841102691867156169633878015853334887726181183
     n+8 = 2^2*9479*90863*15503173813*p533
     p552a and p552b are different 552-digit primes.

     n-1 = 5*53*151*259499*c542
     n+9 = 5*7*31*37*288232349*593280102498871*c524
     n-1 and n+9 are unlikely to have other prime factors below 10^30.

     There is a checked verification for k=9 in
     http://physics.open.ac.uk/~dbroadhu/cert/ifacgg1.zip

k=9: 512 digits. Same as n+1 to n+9 for former 512-digit record for k=10.

k=9: 509 digits. Same as n+1 to n+9 for former 509-digit record for k=10.

k=10: 521 digits. Same as n+2 to n+11 in current record for k=12.

k=10: 515 digits. Same as n+1 to n+10 for former 515-digit record for k=11.

k=10: 512 digits.
    (n   = 4*(x-1)^2*(x^2+x+1)^2*(8*x^6-16*x^3+3)^2-9, where x=2*10^28+2204662)
     n   = 3^3*5*7*307*347*1187*9547*9697*15073*1531843*116110381*250277089*30578866463*111382910297*181158721739*446924830771*1867760574562610834993963*1488459549869389482650623003*8565397930808568846216431951262151*59769847830356822480851338650284237240697374480196117*300623037354997684760214480023188666593095587125093642757*20007803893063496839861142546375474591403195655644157207244713104487*3829797396593923020158058852535774081399736913920378739809259597238377868603562116552089648836044163530241297141741130983038474191260511793881126508518363309739
     n+1 = 2^2*73*127*164881*1930288121*323256465041*90131193051180529*9303255826395894981169*5936198519764907325568788492837144034143764224325971823769414076715761951476514609129427797957422126713145651766928219379641251620723614216506369*3533742054174816074881641068866923118609347812218930248817864339645077320213576784897215431571348169712407642330812093231866486812662973436596195813720944828559128120676691802790771651955258619892556423839559184398326894169807919891967869197534212067400868464481337615037718331079392635959598394921
     n+2 = 29*149*5901418121*p499a
     n+3 = 2*3*499*2124821*13445407369*186826766724947388177434694180263630044871*95925605271645469122449575886666011652340523511711773270825209659498635468887476993851227773969941116968383660751*43778223112892451569864188936071609886439580867801305260814535751043759445135226017605038059205206222938075118226512125698742921604813012867568950280287564892654933848759437580955004560241577043651080237642470767694633379466907843523463211758931466512079866679115169383641194815890749469256483022750032129109031793983093916727712646977883
     n+4 = 239*251*519846829*p499b
     n+5 = 2^5*5*83*331*5413*1645601*13246099*831103267*442781686703*3271127680535279*6263540405958799*37333369881660386459*1730305848698194550273*34830720932848958310122536067257006458277*918030371532084486823140149211743092389268185858411703840428240548268686265325464850510544432796821911496795647262073608062725901351129418695163782977208778237112427886380213*8308098770575618563278140349708230566973798610696444827765518568173848312637469276391944462875073418410679255997789214467944099150573616414945633512750491792050045360588460558085423
     n+6 = 3*11*13*23*41641*2894929*245823731*14734834751929*1248860046909467*1714352630976062017*68696646377151940356904991848691*826402995956294226375591690515963*17205979087403906335627860980115911179419243870279319985508971*4050624313071804187996566630261827216555059616139796924436663460527*12431506263961695057536834021915617377667459367414347205489562416699*1083419113812114581692806946928012991603417612837816992039548006438463*136522321437341357578982495960004506241249989563480223954886877392820440017871099119741773149006344599842089018411
     n+7 = 2*7^2*17*31*191*449*3257*11068417*472514066396281571944993*81654641695766717900650891421256077306657*216432437109204897481725950423278756761076506433*207058905269150904115025146899823757889865604599393*103023928716175825862864924563274619099593728159866663736091990211978293021566631081305017858381298976256350246612433187999055255757621632033668070306853022509953*2359447004608294930877136571811981566820276927752364320206451612966434423077370369933866976913391759307946321794763605121319860302775451881015341532450655411517502262201
     n+8 = 19*2927*5519*180473*1401203*11354621*18811567*113522593*14554641825673780279*179187138283689054518803*2100726463287606221888948966791*1235343327543615764457795495590772030447575628146102891785176718375548949498716110403276999113946325913624500215626384579131612707424323473497173390937204598816818971806525217501534898451*5268468875101207966503221637564275430828482281288806272352342013338174338145138429373064209988535633949696733331592548352583494113390219823457115218230511678697789109397011263873411779303893037684709359210406473
     n+9 = 2^2*3^2*41^2*67^2*4099^2*31189^2*257987^2*2689200539563^2*233437103518022038867^2*741479532797940551572286509299641323^2*97584776774823127592117147231031959014393755760072393^2*8673383825406446118969387417345665128610749211198444515481^2*38250750512834751554907456243913073019849245258367375835003^2
     p499a and p499b are different 499-digit primes.

     n-1 = 2*1483*1933*21841*3791911*2358094721*71075504580403*c471
     n+10= 5*37*272549*c505
     1500 ECM curves at B1=1000000.

     There is a checked verification for k=10 (also former record for k=9) in 
     http://immortaltheory.com/cnt/VERIFY_CONSEC.gp.

k=10: 509 digits.
    (n   = y*(y-23)*(y-41)*(y-64)/55440-4, where y=(13860*(10^60+1898683))^2)
     n   = 2^2*97*p506
     n+1 = 3*19*443*p504
     n+2 = 2*547*11551*2635904987582755423*57944354495692285367*p464
     n+3 = 79*163*p505
     n+4 = 2^4*3^2*5*7*11*17^2*29*199*569*3361*4783*6323*9871*27917*50666557*93548249*251419691*401234363*894724253*3501280801*4455055873*32445623759*156227602720306951640012557918423328611*206195408396375367587226324189737887279*50011980385626978791972075975276821553547719381*777768023292308549684490118627071144703463969328314641*58823529411764705882352941176470588235294117647058823641099*58823529411764705882352941176470588235294117647058823641099*17755515675727712860192365814762364249466181168372134315497429896352598545495965768332329930699664898435392329
     n+5 = 31*37*47*653*919*125925257*7185033209071*112156733297849*328882642025827*79593348847085399*32883211741045463029*81115540365972458569*206921334578038811201*141676523493128094900799*1360783001639169584324399*58956746641484424911415529*233729786242778087790871756339*7748115078873569467096278095337513977*10185312414473538898820214907351318097219*113290015862499223221533450621707149727019427886981*821010301350627818810685153418641068867042983145299511812291*26557022341330289473110636903413510403866099214738127697823875132529407320405318271
     n+6 = 2*167*152063*4590427*p495
     n+7 = 3*89*112223*p501
     n+8 = 2^2*13*16987*218249*356959*49526591*681273809*9106676377*19485931311178781*4967165135717508397*5499182353059262159*1001796858149748467827771*9069406043245243852023611*540526918608745881486206876437*2605643768471675941900168096886934928807*612167007455266691724205830229887098156127313*4307315168277185986198656054312024939833935058255871560498098605127723049*377554735891343947547657524494990782049555797943824283943584385821675669345829*171315516778112833179949573659386968855037125861456391334423927484245873649681869933101158002050587
     n+9 = 5*880133*3463370921120207*232879941578044699*19041760972448845393261*p447
     n was constructed such that n+4, n+5 and n+8 are each the product of known divisors with
     at most 128 digits. Divisors of that size are possible to factor one at a time.

     n-1 = 5^2*41*23689760674499*20201699120271441240691*c470
     n+10= 2*3*23*449*304781745193811*47370240325563932561*432893534978100829771*c449
     2000 ECM curves with B1=3000000, B2=4*10^9.

     There is a checked verification for k=10 (includes former record for k=9) in
     http://physics.open.ac.uk/~dbroadhu/cert/ifac10.zip.

k=11: 521 digits. Same as n+1 to n+11 in current record for k=12.

k=11: 515 digits.
    (n   = y*(2*y-11)^2/9-10 where y=(2*x^3-10*x-3)^2 and x=3*10^28+45140566)
     n   = 4253*226013*230357*416963*2861569*1880854277*42701426129693*13069902194569289*386588385678617711*500005063517754329373345969561796727584786706636720539573*4341248342043895913301460431960445634863440889511346793543879*47352224009928962954139286082311927636952488190447629980094200050881454389562654674388383721111298495341024455450378605701758635972436503172206041376444082433826157671195552288119771345608963121556060908695665231599547748300479567194952011445688867500985464164196239064860297609767831288273189825629275882499448373
     n+1 = 2^5*3^3*5^2*17*647*7793*101027*387371*2876737*3190301*41234059*448017726073*4549647604301*7058560906391*82564644533297*313389057302287*547062257012137*176410881685979243*33529563487769597944543*3905797942598779457796869634480654828557063398268430209*5222735987089480599839721467537544683475254726525559707*126196121207067437991213318947836224809604713521480208688750619923880867*702704165961837523078979683266083025775067212012656945407515732734186659*184430385185788581648109351806679515278594015607579122998192486417070930340664716216441165690089873
     n+2 = 49927*205863965163809*2931670092235410257*533262603377131368593*p456
     n+3 = 2*7*19*29*31*197*12601*554284919*44696195304881*352313237452213*20936955960832581457*64810010594945926073*12486408345477232374233*823966587170283506206259822419*5493295331637810589063791182291419*290061485985760493441318238323975369*913349575170071565559047108472153379766319447*380514144459247521270215147347137364317548994601*39804603645056021960436070859321292255344545643899008580117*6911430939104636279124927560729205170078292892867255581576433921302778449907893288350968461824993673302019230391530510666265516371284599573288697014309779
     n+4 = 3*97*173*p510a
     n+5 = 2^2*3529*p510b
     n+6 = 5*433*1511*5839*54877*70901*122219*304807*716819*1264213*1590521*2004001*6079961*602101170323*117556663309961*3809348476481813575538819198921*12831583161950832470071816910747*6431878224455115560889018029640943*4499242002482501219371685439637499957*1576088009746625511844899685544903239199*82167967255593167594806666269023272477390195236758647209681732983476457238690798651459261400520823548148134390254753813*17479824887805762459776111297664158300407464356342895406325546088030031811326503271187368529128996979561143882697282969022622722284955787
     n+7 = 2*3*421*1931*p508
     n+8 = 151*1067329*802239817*6599032263217*23817728203289561*883765947396190626787314222011033197735982012683064116394694384312468507385219425031573964810769850422257735631417704198370913509927691250034726865938966834569*613596283386608438258179573245611327009544063521439064418060929105150853857219744553356145561979397526887277403972646225035820934179866774672158662530913625446619207540431278244213112656505959897999772260600811020086509981580973246185253854296357393794483907554165864822538249281605736418530475133278726111833
     n+9 = 2^3*89*101*113*179*521*8761*2996893*1385768831*4159235811161*21189480447750425779*746557470021770082838520077807*536425349409413320648023041422658535656199131337160825893487*4890845088036533014507974304316426334045685499764180221896202579301*3764607638136768105709541437596851092368933279685164279140523803173057942543926494529318439716150419*615012230835188011049059046392431382469745444689757743049199070972367899227064135417086557000328620017869735179939215758713862634224557943586663150054707229081347091467892171214006999591590114221
     n+10= 3^2*7^2*11^4*13^2*929^2*4903^2*24497673032853345979^2*82548838381230577645194631472018766442910240852860847257^2*5284274390840591055900162350171249633036536528107034354024855679025659783826864691^2*65289483804720539669247608448854856335421158107620593110415396059725296456406814441995691^2
     p510a and p510b are different 510-digit primes.

     n-1 = 2*11*2377*1861437640330348699939070011*483058876581817566371212578148357*c450
     n+11= 2*5*18289*5453317*4322671468897*105475130567783166422952459096726461*c455
     n-1 and n+11 are unlikely to have other prime factors below 10^35.

     There is a checked verification for k=11 in
     http://physics.open.ac.uk/~dbroadhu/cert/ifac515g.zip

Credited programs and projects

External links

Made by Jens Kruse Andersen, jens.k.a@get2net.dk   home
First version 6 August 2007. Last updated 25 December 2018.