Factorizations of values of the cyclotomic polynomials Phi_n(z) evaluated at z=10

Last update

October 5, 2008 (Sunday).

Sequence and Limitations

Phi_n(10) with 1<=n<=100000.

Introduction

The aim of this project has changed.

From now, we try to find (prime) factors of the sequence Phi_n(10),
where Phi_n(z) denotes the cyclotomic polynomial (with one variable z) of degree phi(n),
where phi denotes Euler's (totient) function.

The only (but sufficient) reference is Makoto Kamada's very frequently updated list Phin10.txt (with factorizations from n=1 up to n=100000).

News

Oct 5, 2008
n=12047: 17360219354095493.
n=12179: 91152033942589363.
n=12213: 476001824023027.
n=12249: 362258253710911.
n=12283: 11856104617558133.
n=12287: 82681508929948889489.
n=12371: 2587188018325133.
n=12387: 13721344509090278881.
n=12401: 52441395948584241769.
n=12481: 9765173790037.
n=12533: 194762888973209372813.
n=12551: 6849751980771547.
n=12555: 9022162027584990961.
n=12565: 6515901828370481.
n=12645: 438444811456452869411521.
n=12671: 571614934165117961.
n=12691: 1554908354976649.
n=12767: 36102468801521711.
n=12771: 1859474567703853.
n=12785: 25368072252511.
n=12809: 81439947641337961957.
n=12869: 3324153360432031.
n=12893: 1495202509882497043.
n=12973: 24142544708028763.
n=12995: 23470393212401.

We found these 25 factors (using P-1) with B1=33e3 and B2=37652736.

n=26022: 634129261225651.
n=26050: 20538053889183643051.
n=26084: 248634245885158801.
n=26110: 13039855896053491.
n=26128: 1196048815705652609.
n=26134: 4399798246489.
n=26162: 6174910773091.
n=26184: 2323617326796529.
n=26320: 7529825406964001.
n=26406: 27128752163606809.
n=26512: 55657995594353.
n=26530: 9013231423922484851.
n=26614: 76507716128760659.
n=26664: 4873036610003761.
n=26722: 17004522387131.
n=26830: 355200655321531.
n=26840: 28208061649447681.
n=26848: 29517302280347521.
n=26854: 510337376320516057447.
n=26922: 4399537484539.
n=26944: 26322615397313.
n=26982: 836091576253125037.
n=26992: 405186065377504321.

The remaining cofactor of Phi_10(26982) is a (probable) prime with 8951 digits.
We found these 23 factors (using P-1) with B1=11e3 and B2=863208-10908642.

n=28080: 14708910528001.
n=28112: 784240843230881.
n=28224: 3277319851009.
n=28224: 5053241557659457.
n=28240: 2741176405311320198401.
n=28244: 2971590627670201.
n=28274: 6686805099731.
n=28314: 5567107683853.
n=28458: 691546985934139.
n=28476: 9357254379283609.
n=28516: 210200636464404881.
n=28548: 6395056251023701.
n=28622: 355899796582969.
n=28692: 3502802710261.
n=28702: 22902273080809.
n=28826: 1652326902715259.
n=28874: 10965391428170579.
n=28926: 102663108839089.
n=28936: 3951531613433.
n=28936: 149162837460001.
n=28944: 1779595985858833.
n=28994: 9886217523407.

We found these 22 factors (using P-1) with B1=1e4 and B2=43570.

n=29380L: 642464608101161.
Sep 29, 2008
n=28182: 112278491463601.
n=28250: 3833641051001.
n=28444: 70933303368929.
n=28490: 237203598266251.
n=28764: 59241957016321.
n=28924: 212756248577881.
n=29620M: 160553731014161.
n=29780L: 20805876631721.
Sep 28, 2008
n=12001: 545550807756193427.
n=12007: 51631948597721.
n=12011: 537596961468606751.
n=12015: 19287916267591.
n=12037: 31821088626656539489.
n=12039: 5698233048799.
n=12039: 128892205484775373.
n=12055: 216700302148081.
n=12067: 2713927724617253.
n=12071: 36320816394529720054951.
n=12085: 14390374818881.
n=12109: 156525700489889.
n=12127: 68761830903613.
n=12129: 5461531682332441.
n=12135: 3516576231567871.
n=12141: 7063026288739129.
n=12147: 3075547705403917.
n=12149: 10101515811889.
n=12151: 6362678289329.
n=12151: 1596576329936881.
n=12155: 26480490344881.
n=12155: 194698265317271.
n=12159: 3701664195391.
n=12181: 106601395158705071.
n=12187: 1540568965253913281.
n=12205: 3914350130527951.
n=12213: 115471155399373.
n=12213: 4205850510759049.
n=12213: 201212562489260401801.
n=12235: 5764945147081.
n=12249: 49507372231921.
n=12253: 57120034011323.
n=12255: 10222937371081.
n=12255: 166681113560791.
n=12261: 2981772960915841.
n=12263: 5851231616958643409797.
n=12265: 743794098204961.
n=12271: 350526278366839.
n=12275: 48135081975925001.
n=12279: 8411002205107.
n=12297: 77888404818907.
n=12303: 19113430594591.
n=12303: 155559068172037.
n=12305: 120122976549551.
n=12315: 4935125316481.
n=12315: 6326328428551.
n=12333: 5659325749117.
n=12333: 88321456693548493.
n=12341: 49856158426297231.
n=12343: 43942534992841.
n=12351: 98309181217503679.
n=12357: 319825386818083.
n=12359: 905966331971839.
n=12361: 2534289448365361.
n=12379: 54325817420597.
n=12379: 809370948931879.
n=12393: 27243864577693.
n=12449: 3624122778319.
n=12451: 10856431756717.
n=12461: 13478759721381601.
n=12465: 3102841549081.
n=12475: 17546345208551.
n=12479: 38712993124129.
n=12485: 19725307267711.
n=12487: 63519400624243.
n=12489: 607565015223853.
n=12497: 88671228435861551.
n=12509: 5594898353507.
n=12519: 400588231547557.
n=12523: 6504779956632813241.
n=12537: 23636808417853.
n=12549: 71121845740963.
n=12575: 21761918840227351.
n=12583: 983557436088679.
n=12607: 457121804040763.
n=12621: 17937796841207209.
n=12629: 28953123351373877.
n=12635: 239832518451191.
n=12635: 33978070818427879711.
n=12639: 119613950553637.
n=12643: 510814030626733.
n=12661: 5367288621883.
n=12663: 588865889409841.
n=12669: 18029737222351.
n=12677: 33918173808175609.
n=12687: 35765548499209.
n=12693: 2830282602923161.
n=12701: 1314550882014169806917.
n=12703: 714107717619551.
n=12709: 4900916767121.
n=12709: 734046836872597.
n=12727: 1620544462903081.
n=12735: 99712055237401.
n=12745: 23236621065191.
n=12753: 248711688261973.
n=12767: 2500048973321239.
n=12783: 16069961267329399.
n=12801: 19098628629403.
n=12817: 82795845180099211441.
n=12819: 734207584796758027.
n=12837: 105303104635609.
n=12839: 402859374450839.
n=12849: 75146990413249.
n=12855: 31170363433441.
n=12855: 33981691489231.
n=12863: 52582174231283.
n=12867: 15424883815042711.
n=12879: 85323004213591.
n=12895: 266765266995711361.
n=12897: 26180653217454163.
n=12907: 23502032953837.
n=12913: 1486216200643157.
n=12917: 63398460112907.
n=12929: 6224478571039.
n=12935: 7406225675551.
n=12941: 1426466054162563.
n=12949: 8132387881997081.
n=12975: 4936694824170601.
n=12979: 2002436134202920721.
n=12997: 18530056393310532335519.

n=24042: 41085154681213.

n=28220M: 15147240943721.
n=28364: 6273600773749.
n=28372: 49425320430169.
n=28784: 170191825509889.
n=28900M: 1361926259831701601.
Sep 22, 2008
n=27004: 8163387160549.
n=27006: 17113154059219.
n=27014: 11138989823209.
n=27038: 721630336940738413.
n=27046: 5186014947517.
n=27048: 32602019588918641.
n=27052: 72674529950861.
n=27078: 7596134340134133121.
n=27098: 22208306403131.
n=27104: 63729682442273.
n=27132: 219471713899201.
n=27146: 94135109443232453.
n=27196: 112930420680169.
n=27214: 1122717597672001.
n=27228: 27497352581581.
n=27266: 1079337846143371.
n=27266: 3861526291174064729.
n=27276: 7394398621369.
n=27316: 7979335844509.
n=27318: 53157744098299.
n=27406: 21334619518493.
n=27414: 47668387244630077.
n=27428: 5445981076841.
n=27428: 64157320824161.
n=27428: 33172233744987581.
n=27440: 177951287446606241.
n=27444: 153421303309801.
n=27476: 4329584331006399161.
n=27480: 4227448179601.
n=27522: 11666613395053.
n=27534: 7238816770771.
n=27544: 5872830042641.
n=27572: 837964473137649969469.
n=27590: 5175887421161.
n=27594: 22039039553077.
n=27600: 30929726596801.
n=27618: 10515579350449.
n=27642: 29089000493245489.
n=27646: 120739960938703.
n=27664: 61150972205233.
n=27674: 6700166057752733.
n=27686: 41867567366977798997.
n=27690: 12457522355851.
n=27696: 1120721185548780545137.
n=27698: 4353709215967.
n=27706: 7082988059491.
n=27722: 9287329020877.
n=27724: 20729313059501269.
n=27744: 153768066828289.
n=27768: 9459321442987598209.
n=27770: 8422683876881.
n=27782: 31218539469059.
n=27812: 72363860504855401.
n=27818: 1031649648733481051.
n=27880: 13507867527601.
n=27884: 82572319585649.
n=27890: 15743429879347201.
n=27930: 14603050264531.
n=27932: 348043906626341.
n=27948: 31650308032141.
Sep 16, 2008
As an exercise (in using Msieve),
we (completely) factored the 167-digit number
Phi_83(246)/9411869/17465260836276023478023.
Phi_83(246): 1927587513220160433819733.
Phi_83(246): 6784752303096857873687616123403.
Phi_83(246): 12418328557665161077248671216019960137563831459.
The remaining cofactor (428534319037059927862695160849220290484528222711342816108777213029) is prime.

Phi_174(820): 3068190162788155006304300018154130382176927.
This factor should finish the factorization of Phi_174(820).

Phi_116(267): 111296547209489918497436669279221309612846013.
This factor should finish the factorization of Phi_116(267).

Phi_276(615): 2765700964974038938616840308837.
Phi_276(700): 521887882245756860857.

Any of these factorizations should be new (at the best of our knowledge).

Efforts

n=12001..12999, odd: P-1 with B1=33e3 and B2=37652736.
n=24001..25000: P-1 with B1=1e4 and B2=863208.
n=25001..26000: P-1 with B1=1e4 and B2=863208.
n=26002..27000, even: P-1 with B1=11e3 and B2=10908642.
n=27002..28000, even: P-1 with B1=1e4 and B2=863208.
n=28002..29000, even: P-1 with B1=4e3 and B2=43570.

Maintained by Alfred Reich.