Factorization of 18 consecutive 303-digit numbers N+k, k = 0 to 17 Discovered by David Broadhurst, December 27, 2009 ************************************************************ Formula for N: x = 9*(2^123 + 63841119) N = (x-2)*(2*x+9)*(2*x^2+3*x-8)*(2*x^2+5*x+5)*(2*x^2+7*x-3)/216 Decimal Expansion of N: 521341191752634055766740682853557803037002859593403051242048475994621208384298812203119069470439236491133248120157475726969118888901287189394696407256057102465221719754746450905482033403542345858172338421109598318109819318029490667316261589304960914814250360741837950297146207553188447286907179479611040 Decimal Log of N: 302.71712204037708897 Prime factors of N: 5 x p1 = 2 p1 = 5 p2 = 31 p2 = 61 p2 = 67 p3 = 127 2 x p3 = 137 p3 = 367 p4 = 1093 p5 = 14843 p5 = 63443 p11 = 27794152609 p12 = 476489052779 p14 = 37329968673889 p16 = 3570689780842673 p17 = 67021429861901987 p19 = 1189940881923530689 p22 = 3793038463420088582879 p34 = 5099068447786986139329431953793989 p37 = 4253529586511730793292182592922639091 p49 = 1247938418976429177194252870942752961878598228407 p49 = 1976165708577417315332451629720246636620404182563 Prime factors of N+1: p8 = 37090393 p17 = 13607868187049611 p17 = 35795338430018551 p23 = 30626694703266864141733 p240 = 942201756024301406105251331672635304101058637337342747709515926557605912104375935454785161515792426418599159420683749641213239877861786064310499129835659857200971495908140445229082812087221791238418769339423233603197170371053849170308533849 Prime factors of N+2: p1 = 2 p1 = 3 p11 = 12255821987 p18 = 924815196911816653 p23 = 77063469250364238854683 p251 = 99477493011438039745124732342422767955806205109555903949502900017680935099967555028147875627805864118290216829799813665554377512621456939071052636356085151110295624894877886709068588236445832024130286100460564097661774006177224725738532167883308106039 Prime factors of N+3: p1 = 7 p2 = 11 p3 = 101 p3 = 701 p11 = 10956299843 p11 = 12992500051 p12 = 845252065961 p14 = 62344405116953 p16 = 3926885974951393 p18 = 336759436885389653 p101 = 41827321408446414865620149519510469805335466541655886006605612312184218289156276291154155470299361539 p117 = 230474797362078527686276487291551259767718914858004558921522786393664431055994745914544894064248605620192451955503081 Prime factors of N+4: 2 x p1 = 2 p2 = 59 p2 = 89 p5 = 10139 p8 = 57650393 p8 = 68792951 p24 = 915637968240141024382421 p26 = 17429980721302463660682247 p29 = 53099629972946910472577102467 p30 = 173016782735821807795582949767 p35 = 17374116368233106666319480778234093 p51 = 103014867632994291246210208672968735911833990892809 p88 = 2352207522232565838674357655486051602440217895012370749415664981032523924946869838703973 Prime factors of N+5: p1 = 3 p1 = 5 p2 = 79 p3 = 991 p297 = 443945885758862722959581961581284559377863096640569387135739355460427568269956028052573643355975285153838766723428558057938423779331525662945153135396677355665309915615856166175309458888257904139936507403006466057904958396053503188882441202301694929312547408313503344699039207341336541350557702427 Prime factors of N+6: p1 = 2 p3 = 167 p301 = 1560901771714473220858505038483705997116775028722763626473199029924015593964966503602152902606105498476446850659154118943021314038626608351481126967832506294806053053157923505705036028154318400772971073117094605742843770413261948105737310147619643457527695690843826198494449723213139063733255028382069 Prime factors of N+7: p303 = 521341191752634055766740682853557803037002859593403051242048475994621208384298812203119069470439236491133248120157475726969118888901287189394696407256057102465221719754746450905482033403542345858172338421109598318109819318029490667316261589304960914814250360741837950297146207553188447286907179479611047 Prime factors of N+8: 3 x p1 = 2 2 x p1 = 3 p2 = 17 p2 = 19 p2 = 97 p3 = 571 p4 = 1879 p10 = 5604570953 p15 = 136343229864379 p18 = 164594447326725833 p18 = 473886220150299667 p19 = 2198225861170227319 p58 = 1371395920627640779933298159039238396871538437980374608021 p71 = 33965653755659856634962651874114717627795799844203065344750819625447681 p86 = 35294907249794919921010634808783401482706946209704380631021053235559507333175109017737 Prime factors of N+9: p2 = 47 p2 = 73 p4 = 2897 p296 = 52450885809935348124602983081077330626553228874481964049690141269631808217799638577573446261048272481108483275058810245412028754145036839926839804356053222473003381296136401661100085084203263354192206836860813341826273344411855586173201977634021235931586667434822921097096314527645655133739913407 Prime factors of N+10: p1 = 2 2 x p1 = 5 p1 = 7 3 x p2 = 13 p2 = 23 p2 = 37 p3 = 229 p3 = 449 p3 = 487 p3 = 883 p4 = 1213 p6 = 117353 p11 = 89687608357 p28 = 1139052822168391395318313531 p33 = 127995606592011373381727503535617 p35 = 66048596063846751448636375666500607 p57 = 144238439425142847822185131778826833524763893459682944181 p115 = 1016136186566902950487292533601142453337659375650957772480057841547153652919857530532279670522279625116051866542077 Prime factors of N+11: p1 = 3 p6 = 125621 p7 = 1722883 p7 = 5031809 p10 = 1458162773 p276 = 109434082112857156325053221326326374762994232725835386966742087615959919987199078357857099955309779887558040085468492619389233846143802405816760130034309753168441213676080877801501400755824580101162175942220213622478824396501663754992312492199442550718867101670434565183881867 Prime factors of N+12: 2 x p1 = 2 p5 = 21839 p6 = 219749 p12 = 117208647991 p53 = 28357232254275817248423317326253740714599966942376919 p64 = 1085354970341538846716082499948829244132306443785398119368051033 p71 = 41680895857597278552165489917866599494531786052396535716431626637953957 p96 = 180621752826093085887810177503344416313467187488754965978200872603027411174173709410724797866117 Prime factors of N+13: p3 = 359 p3 = 479 p4 = 2153 p7 = 7265911 p11 = 21512884867 p19 = 2858777152850361559 p53 = 21788194043760237961997150009186546891903780196450053 p61 = 8000588567727090488546622009895596947183330276157220193875589 p146 = 18077394989497099734661151013889572047300794586297821113980772282067387680992522454540504939337641465206207910922648325602284099187317316510527031 Prime factors of N+14: p1 = 2 p1 = 3 p2 = 11 p2 = 29 p2 = 53 p6 = 152389 p11 = 52970372003 p13 = 2708177799253 p17 = 87983394132360053 p20 = 97498870451759261363 p22 = 1952473503578915266687 p23 = 34276804077887529565543 p25 = 2141760271268170396157261 p38 = 11963051962064242856134263542594922443 p128 = 15982393058345528623714538970845053489498787266629839518465535586618814341796316176833772093483478647066267566611828359350069181 Prime factors of N+15: p1 = 5 p3 = 103 p3 = 107 p3 = 149 p3 = 151 p5 = 25931 p7 = 1005581 p9 = 128443787 p11 = 12326610809 p13 = 9131359354069 p19 = 1401135788929920143 p28 = 1000144156045016508931360243 p33 = 440769109203489208487649672110629 p63 = 334354569736274739561425338719692676876970791692618251383721757 p112 = 5401047946589175641831344745113630631098190906371295032040451816833769174030621378871777224763858006606966651001 Prime factors of N+16: 4 x p1 = 2 p29 = 59884477891211209977652573873 p273 = 544111356263852622557455181686109755982053686281013821678500136133049604129381475819195176355964912499856158603148078527372218004204413707392209290380949530271802433781267532925248205069266776672307212039564980606149098963453828684016704024145060378452985252853211320949467 Prime factors of N+17: 3 x p1 = 3 p1 = 7 p7 = 2039351 p7 = 6934549 p16 = 1330188696705607 p273 = 146634709005136852368414830976024009931854668137818099745734790434483085409254871319200344889908751708399499999235562580038333779653249991057177146951868845532062481079728410692721821435444662618299285619174111332843796554046924833436954713939738918155967461959198886521841 ************************************************************ Extending the sequence requires factorization of one of the following: c279 = (N-1)/(3^2*7660729*75495982961295157) c276 = (N+18)/(2*811*2777*562633*398311899246763) which are unlikely to have a prime divisor less than 10^35.