References : Encyclopedia of Integer Sequences :
for k=2 see
M0728,
for k=3 see
M1551,
and for k=4 see
M1957
Remarks :
1.- a_2(44) is not an integer.
2.- a_2(n) can be defined by the following formula for n>1 :
a(2)=2 and a(n)=a(n-1)+a(n-1)*(a(n-1)-1)/(n-1); or
n!a(n+1)=a(2)(a(2)+1)(a(3)+2)...(a(n)+n-1).
With the programs "factorize" of Torbjörn Granlund and "ecm3" of Paul Zimmermann (which both use GMP of Torbjörn Granlund) and the computer algebra systems Maple and MuPAD we obtained the following results :
For k=2 :
4a(13)/a(12) has a prime factor(test by Maple and factorize, proof by MuPAD) 4341757003926539859424403 (25 digits)
13a(14)/a(13) has a factor with 82 digits :
767236744795310780646500690794493256107993822414657298102831365
4438300542454121313.
By ecm we had prouved that c82=c32*c51.
42361466846825704423324431958777 (32 digits) and
181116661415326453070118939819157469149663699744169 (51 digits).
2a(15)/a(14) has a prime factor (test by Maple and factorize, proof by MuPAD) 1981106063293982628008437003983046432365598300533234401842830961390514 6361783881801419290177040633550140341312347689455443940660090563736863 6005456486609181. (155 digits). The two others factors are 587 and 309810101.
a(16) has 667 digits; and
a(16)/a(15) has the following factors 2, 29, 577, 45263, 150366473,
109707166083128153, 350688642212729, 5749883908284510629 and a
composite factor with 264 digits:
5464532019644563659262672585775784162691667270507473263634151058759327
3911488527887438145746280725988616282314255173437043670009902916819829
1221207068903553660578145161718271823260844081228186556370760889545098
676655117099578422149866619451903957255910123187944973.
Found probable prime factor of : 3846713236546690610415342293815733351321939
Probable prime cofactor 142057171502345657076625647229490504128109254617994018601368869000955009469652612327214995047489211990967218402382945210890494158135014137588712589678243816844503705974690557765334302109831140450669209774689829358367352607 has 222 digits
For k=3.
a(8,3)/a(7,3) has the factors 3 5 102101 200333669 and the probable prime cofactor 2322056429151918046562159647871502333971904980317327740727 has 58 digits (prime proof by MuPAD).
a(9,3)/a(8,3)/79 =
4005206103210066326877329293853446378288405202069518204422144310496703589
2932471094867914789999097350017840328123688347057158954213116929825460744
35931619596021890750809876243062847156310099840957884070120232417551
(c214)
Found probable prime factor of 30 digits: 226606153128119565713464491223
Found probable prime factor of
Composite cofactor 93348065704926068752981863730537869115747929901911028
784587487025420048265413776169652204483800494135264946788734957422831182
435858205257208074754791 has 149 digits
For k=4.
3a(7,4)/a(6,4)/379 = 147419592855542737789440527865055084545487095492014413250889447284402837791 12958872916610421681465870692267621675809786507485857789764050033536903097 (c149) by ecm a prime factor of 28 digits: 1367919568960169498955547991 and a composite cofactor of 122 digits: 10776919652345088930014977534732438432923444510241640662127136919959539939 309242894484157283897657224612382419685688934767. (by ecm : probably no factor having less than 35 digits).
For k=5.
using ecm, a(6,5)/a(5,5)/397 has the factors 885517, 3987598108577, 3973924970331061458216027001 (28 digits) and 5029777094143060295415476053433593 (34 digits).
And so on ... . For exemple
u:=a(4,19)/a(3,19):
ifactor(u,easy);(11) (457) _c89 (174763)
c89:=12967874071584619221793449567793026696204380002101839495484270599026435418432063004747809
using the program "factorize", 12503926697 is a divisor so it remains
c79:=1037104134231363626314887182339103782751836777030798639164820641224720018720697
Found probable prime factor of 12 digits: 393263370163
Found probable prime factor of 27 digits: 142802267787510351875763907
Probable prime cofactor 18467315313726987167487552803689421848417 has 41 digits
u:=3*a(4,20)/a(3,20);
ifactor(u,easy);(17) (173) _c101 (61681)
c101:=25893916474174674724660293283304598978932084171864476113151632827121087083340324212276289500587239791.
using the program "factorize", 2622679 is a divisor so it remains
c94:=9873078815278070524322760537337813349987582991233191752841896712148565296530884722177700549929.
Using ecm we found
a probable prime factor of 37 digits: 5737599236330868458465441580907624633
Probable prime cofactor 1720768288025602102803876034381707450609211313851140220913 has 58 digits
A Maple program for a :
a:=proc(n,k)
options remember;
local i;
if n=2 then 2 else
a(n-1,k)+a(n-1,k)*(a(n-1,k)^(k-1)-1)/(n-1) fi;
end:
Now generalized Sloane's sequences.
Consider the following Maple program:
m:=proc(x,y,n,k)
local i,s;
options remember;
if n=1 then s:=x else
s:=0;for i from 1 to n-1 do
s:=s+m(x,y,i,k)^(k) od; s:=(y+s)/(n-1) fi;
end:
Remarks
i) - for x=y=1, we obtain sloane'sequences.
ii) - for x=2, y=-1 and n=2 we obtain Mersenne'sequence.
seq(m(2,-1,2,ithprime(j)),j=1..7);
iii) - more generally for x a prime number, y=-1 and n=2
we obtain the Cunningham numbers.
iv) - in all cases we have (n-1)*m(x,y,n,k)/m(x,y,n-1,k)=(n-2)+m(x,y,n-1,k)^(k-1).
In the next we study numbers obtained with y=-1 and n=3.
For this, we consider the number u:=2*m(y,-1,3,n)/m(y,-1,2,n); this
number is nothing else but (y^n-1)^(n-1)+1.
Using the following Maple program:
truc:=proc(y,n)
local a,i,A;
global u,B,C;
u:=2*m(y,-1,3,n)/m(y,-1,2,n);
a:=factor(x^(n-1)+1);print(a);
if type(a,`*`) then A:=[op(a)];
C:=seq(subs(x=y^n-1,A[i]),i=1..nops(A));
B:=seq(ifactor(subs(x=y^n-1,A[i]),easy),i=1..nops(A));print(B)
else print(ifactor(u,easy)) fi;
end:
we obtain quickly a first decomposition. After we use the
GMP-ECM program of P. Zimmermann (Inria) in order to "break" the
remaining factors.
Some results
u:=m(2,-1,3,17)/m(2,-1,2,17);
u:=(193) (19657057193125337947134113691145024644399675465145762452378984383645863809158977)
u:=m(2,-1,3,24)/m(2,-1,2,24)/2^23;
it is the following number : (2^24-1)^23+1;
_C142 (75533) (60169) (4877) (6073)
C142:=6527870805502626093342093333531062515959139059493140371023560834156184852397361179422015804361601344180736447299730326353342127573568648305983
Found probable prime factor of 21 digits: 805929128496354948959
Found probable prime factor of 26 digits: 26973262313379844902618329
Found probable prime factor of
Probable prime cofactor 1611437098160275782305046325915346139178682395307015083 has 55 digits
u:=2*m(5,-1,3,ithprime(7))/m(5,-1,2,ithprime(7)); (the number
(5^17-1)^16+1 with 191 digits);
Found probable prime factor of 8 digits: 31805729
Found probable prime factor of 15 digits: 760311658174433
Found probable prime factor of 22 digits: 3182040172353218823553
Composite cofactor
c147:=171253129747478510713753717408085429033184002526143131450701370547243399503959477674693468320504583627487732206638506032072291600048383700965553537
Found probable prime factor of 40 digits: 1653107399970694443954636520243006311041
Probable prime cofactor 103594678573524267311786002558304608119989749780241530250096168997516092697753423192276214779062340224576257 has 108 digits
u:=2*m(11,-1,3,ithprime(6))/m(11,-1,2,ithprime(6));
c116 (161761) (10273) (16396150647649) (1143490991549960064206017) with
c116:=91984176788618832284771645726718641129860744454269135826479950497341758143462782269919825574598354628554043107655649 has 116 digits
Found probable prime factor of 30 digits: 629007680590589341415243312617
Found probable prime factor of 33 digits: 102952654608074247614634957635497
Probable prime cofactor 1420429319844148750619507605173129268966835456856010001 has 55 digits
u:=2*m(13,-1,3,ithprime(5))/m(13,-1,2,ithprime(5)); u=(13^11-1)^10+1;
(461) _c120 with
c120:=741426028598991006882450541864025606517046715161416984155851156926876160647712933166753495440274989870276140194534314757
Found probable prime factor of 25 digits: 3211838877951270784369297
Found probable prime factor of 37 digits: 3881525110251159763758055822839638161
Probable prime cofactor 59471881386608397551050153749657074839381435504844533086821 has 59 digits
u:=2*m(51,-1,3,ithprime(5))/m(51,-1,2,ithprime(5));(u:=(51^11-1)^10+1);
c184:=1638045906218337001999922154500716763628663653138827356807939848712356192308797090394748672085516796150310218630275937201334612737433647094371329077195523796219909245299391541816146197
Found probable prime factor of 10 digits: 9331405541
Found probable prime factor of 8 digits: 13371161
Found probable prime factor of 10 digits: 4700700181
Found probable prime factor of 8 digits: 46491281
Found COMPOSITE factor of 23 digits: 95105034595562531514517
Found probable prime factor of
Composite cofactor 115800307226543655830683770691765088716497845384771864365033458656017766936041772186741 has 87 digits
Found probable prime factor of
Probable prime cofactor 68862914807458133446509499353136819156781929561 has 47 digits
u:=2*m(69,-1,3,ithprime(5))/m(69,-1,2,ithprime(5));
u is the following number : (69^11-1)^10+1 .
(5)^3 (41) _c174 (41849) (162529) (6221) (73981289) (9041)
c174:=129387284669038020478301416368435868500743068128624978049200235413449208019027081629626805297732286340246342113754678273783112719363955137678174950130646818183061325933510577
Found probable prime factor of 8 digits: 10305341
Found probable prime factor of 11 digits: 54032967061
Found probable prime factor of 18 digits: 250487279711579497
Found probable prime factor of
Probable prime cofactor 44659065950583553047446800850717580697272485152294229153930180542587596139736850384636078197892701
u:=2*m(2,2,3,23)/m(2,2,2,23); u is equal to (2^23+2)^22+1;
u=(29) (137) (233) (76016089) _C128 (2667061) (30493)
C128:=36607238921640409315769706444177090870694725111057761156426895238326326206050787607721185782382630065290696756267607301601023437
Found probable prime factor of 9 digits: 327055081
Composite cofactor
c120:=111929889025758353271905616547131676791453746321885643343443104124489434873053654248903675593355573232298130100805886277
Found probable prime factor of
Probable prime cofactor 88392050215868058799037494572992870726563779355404050580111740239099164121 has 74 digits
Let a:=(85^16-1)^15+1; then we have the following decomposition:
(85)^16, (5) (11) (131) (211) _c113 (170341), _c56 (1414801), (61) _c246
with c113:=11737500203147420239963595635987345592406927088529592972728608594660349069539389297557654121150787531420758784211
Found probable prime factor of 39 digits: 854172125440244024574143792546215157111
Probable prime cofactor
13741375834639723977571756190649403432046106525094401091142930699732756101
has 74 digits
Let a:=(145^16-1)^15+1; then we have a c132 as factor:
c132:=110583754069328913001369606987037359784128770454446588842509915274065823247673093524874097358839594316084753430050659903066250845051
Found probable prime factor of 10 digits: 1140983341
Found probable prime factor of 13 digits: 1834416398341
Found probable prime factor of 37 digits: 3663292634598435280776373346606398201
Probable prime cofactor
14422564466706081774431019116494250667603184364917733570423849636477817971
has 74 digits
Let a:=(2^24+3)^23+1;then we have:
a=(2)^2 (5) (47) (397) _C158 (2113) with
C158:=18695691592111634675049924500515913593912383944935776502208961413362307167339738706545910934051108012790603354064765426902119106929945407817046611393886267229
Found probable prime factor of 22 digits: 3881610165914554487927
Found probable prime factor of
Composite cofactor 2104979522275064751801002064364611490954357893597959005287936026014940556817011596032653480208333 has 97 digits
With a:=(126^16-1)^15+1, we have:
a/126^16 =(211) _C123 (31091) (54181)* (3) (37) _c66* _c260 (13711) (66751)
where C123:=746364071346188970006545017569506961622533916489526691026093861800075581354496997181867960366728039506757303588105979602521.
Found probable prime factor of 14 digits: 57675946866011.
Found probable prime factor of 38 digits:79587270087587482577035759123663269311
Probable prime cofactor 162596945845558007247159917722253704539113850700227047133781869576032101 has 72 digits
Let a:=(161^16-1)^15+1, we have:
a/161^16 =(1061) _C139, _c71, (31) _c272 (4105774741) with
C139:=1626075409267221315812488917538263039576412680222494474460098121723611160466738930577444439474360588451338069596302441128775643029413451181
Found probable prime factor of 13 digits: 3628603435241
Found probable prime factor of 21 digits: 674077527905499528751
Found probable prime factor of
Probable prime cofactor 357708185838230404928676265758344716166377734162318590531540069741 has 66 digits
Let a:=(164^16-1)^15+1, we have:
a/164^16 = (61) _C140, (7) (37) _c69, _c277 (6825331) with
C140:=92188434496789116188064684747924334619791349354361569029345973083986000702528175161342210813404272198181549707741507983471428960609516843881
Found probable prime factor of 14 digits: 70927005646501
Found probable prime factor of 38 digits: 46961608604461642890352350340359181271
Composite cofactor
27677180747575033942576844936191672338896395335265479076866507694770064046977911747275411
has 89 digits
With a:=(157^16-1)^15+1), we obtain :
a/157^16 = (11) (61) _C138,
(7) (2414409736438666357878591640998090747028708203657808969348147) (3583)
(306643),(691) _c279 with
C138:=513872330891787143201191334033892880590488410143801919388153605925799428294243502539821270564590308062486476759274550414264672622419958431
Found probable prime factor of 37 digits: 2263610427177784317024985644556866311
Found probable prime factor of
Probable prime cofactor 2878548357774618807546070155248997753093003520569498761 has 55 digits
Consider now a:=(29^13-1)^12+1, we have the following factors :
(41) (17431201127298507088197235193693424852311881779136459756933624079729)
(2393) (6481), _C149 (1801), where
C149:=68214714949060003135201493005452929023163897159372188748607052768979581200664506171488928081983624194029810312328065478959033819876262934587221896761
Found probable prime factor of 13 digits: 2062310869033
Found probable prime factor of 13 digits: 4757829638377
Found probable prime factor of
Probable prime cofactor 1457966334839320723355676006223554316355262813386730912314268774696429852400152337 has 82 digits
Let us put a:=(31^13-1)^12+1, then we have :
a/31^13 =(89) _c72 (4241), (73) (97) (1657) _C149 where
C149:=10769587173175443993382348404905602844109180357600801468810084182408406036287265857651905782087801310886850692928171856980886555947550738728557313553
Found probable prime factor of 15 digits: 160742567366833
Found probable prime factor of 26 digits: 11869851619120501313299177
Found probable prime factor of 39 digits: 444269266200529949219666962767780946153
Probable prime cofactor 12705056328361003175501437672213884921106996827542613782774230234746961 has 71 digits
With a:=(170^16-1)^15+1, the following factors occur :
(5) (31) _C137 (11681), (37) _c70, (181) _c284, where
C137:=30968434102256265361512495715353046686740762755823706357118853369039449941757625237608810116732878769742560979854692308884928599241668991
Found probable prime factor of 11 digits: 81211564121
Found probable prime factor of 39 digits: 106172660801126960209642467465326880091
Probable prime cofactor 3591605845466308650912925084689575266796283006873015282695593063733827237291830051832181 has 88 digits
Let us consider now a:=(200^16-1)^15+1; and its first
decomposition:(5) (11) _C146,(7) (483681797208122361908161404729844879311229173412596338857907429311) (37) (342847), (31) _c289 (58171) then
C146:=33539534679471912029090909090909090883502274844485818181818181818181818189627213265454545454545454545454545453353890909090909090909090909090909091
Found probable prime factor of 34 digits: 8809578323210404673856723643924231
Found probable prime factor of
Probable prime cofactor
740731871974236982345915976084962663857758439472312448669135896463431
has 69 digits
Now put a:=(6^36-1)^35+1; then
(29) _C167, _C108 (12491), _c669 (3221) where
C108:=906114013190592113019006293327541308721526353814762906527646686965246561880131472978950372805893830540626031
Found probable prime factor of 10 digits: 6823374931
Found probable prime factor of
Probable prime cofactor 534053835015416959393612652307751487157919626136171591 has 54 digits
Soit a:=(93^11-1)^10+1; ce nombre se decompose en :
_C174, (20259320183077556583942113108294437507288337), ou
C174:=168461038157741731045423641778835726198863763532303342773635688869696200690522004631071234670465460604690980180837507510901055392672377317417256220637715826435408590524302321
Found probable prime factor of 11 digits: 66732472921
Found probable prime factor of 13 digits: 1714604982221
Found probable prime factor of 19 digits: 1044876997351715741
Found probable prime factor of 30 digits: 112112946338005205877938871341
Found probable prime factor of
Probable prime cofactor
1314802647081173878518896130595368669706310641060252948943505881 has
64 digits
Prenons a:=(2^31-1)^30+1;
a= (2) (5) (733) (368140581013) (1709), (5) _c74, _c38, _c150, avec
c150:=204586911469217587909732290282635571005434969041506395971841045832567232635129768477684486843939097549979379629542655588229202929516797589944491048961
Found probable prime factor of 39 digits: 188393578481774051844552516742742458801
Probable prime cofactor
1085954803332164232202975450223967864188905233751283062530440322701979522912269823374656295252140420911533582161
has 112 digits
If we take a:=(5^56-1)^55+1; then
a=(5)^56, (23) _c391, (5) (191) _C151 (2411), _c1566 where
C151:=1610943798466201543854078606364650462584967657054959717447882361420202749050087897266772802487694623835762691157782881849286504126849517042325333227001
Found probable prime factor of 16 digits: 1981525394257921
Found probable prime factor of 16 digits: 4980701898895511
Found probable prime factor of
Probable prime cofactor 158424987154273737601357830337472801886016458227746077625772808625502816216988521 has 81 digits
for a:=(183^16-1)^15+1; a/183^16= (31) (1291) _C141, (3) _c59 (3901411) (18619339),(331) _c288
C141:=156524313124332350511308450825790333134170139230810465882040832603072671360026559418273442607618769835645871605542955653133540197467122661501
Found probable prime factor of 19 digits: 2928412791407446801
Found probable prime factor of
Probable prime cofactor
208339298128693453335360221558306913115776351386610707762435897888460173081
has 75 digits
Soit a:=(3^46-1)^45+1; on a les diviseurs:
(3)^46, (11) (31) (41) (431) _c73 (92831) (2971),
(3) (7)^2 (43) (79) (54487360321244902700971678251914929) (2887), _c176,
(3) (19) _c127 (5437), _c518 (393301) (6301), avec
c127:=1563983309151330303811046219674930136262244798831844879236976742490342288181555848395958460269956097616637328634265914466646717
Found probable prime factor of
Probable prime cofactor 5193359518886217275320459717646637800029165680258036646342462584020937889678643457211 has 85 digits
Soit a:=(2^132-1)^65+1; on a la factorisation partielle:
2^132, (79) _c471 (21191), _C155 (12041),_c1903 (23011), avec
C155:=72975176521610999378596736341036320223376859507635655964473369988746664418584190518991524722727920133701237249171446829345188574334147844278787388800781341
Found probable prime factor of 11 digits: 13260244331
Found probable prime factor of 25 digits: 1683217799462864982926171
Found probable prime factor of
Probable prime cofactor 1962536492898291983817341937978044482867411196792044091015183928437185977281 has 76 digits
Soit a=truc(3,77). a=(3^77-1)^76+1= (97)
_c141 (47137), etc ; avec
c141:=196432245228461791212036261411708979987861119137457534836319956996065797245998758904558285725440782905464670227832426716989139796486248850833
Found probable prime factor of
Probable prime cofactor 44390689847354504906458786470486026986049908028428143170157136838609748537945461388737367702897215473 has 101 digits
Soit a=truc(14,82)=(14^82-1)^81+1=_c185 (3499), (109) (181) _c560, etc .
c185:=26366586525802562398492592060103767915510476947777079045816534009495555756976403631028545721577775716193715100958774756427627897154452436721857385576346382554651291951940590129879944969
Found probable prime factor of 14 digits: 11592002419693
Found probable prime factor of 18 digits: 893860454611578397
Found probable prime factor of 29 digits: 44136216255057790373635044631
Found probable prime factor of
Probable prime cofactor 24433892579707256096924235169935496667537835278698582874297839768309611923257 has 77 digits
Soit a=truc(8,82)=(2^328-1)^81+1 = _C142 (4467601), etc .
C142:=2862091324201580261921112080786267437120337736596379541037596443063183718032681374987453894577441001049111313842575605829534926781289650970707
Found probable prime factor of 11 digits: 20909852791
Found probable prime factor of 9 digits: 140805079
Found probable prime factor of 8 digits: 98025703
Found probable prime factor of 11 digits: 39034430299
Found probable prime factor of
Probable prime cofactor
411807357021306002887956399375017097940199928102052534412599 has 60
digits
Soit a=truc(9,88)=(3^176-1)^88+1 = (3) (163) _C146 (828864739)
(224209) (208657),etc .
C146:=46644789444518298151297957603375735981057094980948424036021047572081302781050802594121733577451620746146417930304268470963831745712860384927872947
Found probable prime factor of 9 digits: 381350269
Found probable prime factor of 32 digits: 1477098223545681576046632850401
Found probable prime factor of
Probable prime cofactor 346766647262795204442492896925332832605952716142676617898811821 has 63 digits
Soit a=truc(4,94)=(4^94-1)^93+1 = ..., (7) (19) _c112, ...
c112:=1157248772215533341526052634520195424256336327257561286056361387480825925012720063413425113572368496812510406887
Found probable prime factor of
Probable prime cofactor 2709442312606258048513738381295866595738445465869570143444780246474219 has 70 digits
Soit a=truc(6,43)=(6^43-1)^42+1 =(2) (17) (373) _c63, (281) (1429)
_c387 (2470964693),(13) (733) _c127 (7177), _c804
c127:=1016306709671193366455664959650795243281414526493491827044789465193046042487698662851664697620567076330351101340248184365348097
Found probable prime factor of
Probable prime cofactor 2099541536720217916793625797492307860357351911231978329066474541503922681 has 73 digits
Soit a=truc(2,106)=(2^106-1)^105+1;(29) (1303) _c187, (11) _c119 (88241011), _c761 (314161), (37) (967) _c60, _c376 (20591803), _c256
c119:=44632889236252921590055755601639090758747939637051261603504635083793806123527050694289502186296462368212523219360335701
Found probable prime factor of 25 digits: 8183908962196473026410871
Probable prime cofactor 5453737259593603153452780353429669215977523202978062215080688974659422772564617248370990659731 has 94 digits
c187:=7546313245332712339298521326282679343981074803446066121383958122243000176722655909418568161596015608619696047411218156801295218739286189139913118666146671830007153213494514895233801796101
Found probable prime factor of 10 digits: 7414936579
Found probable prime factor of 22 digits: 5423369798124337271213
Found probable prime factor of 28 digits: 1636346644000094117326789387
Found probable prime factor of
Probable prime cofactor 96336990253332641673156109879707341262499156074291492850929697230769103123829029716969 has 86 digits
Soit a=truc(2000,9)=(2000^9-1)^8+1= (2) (17) (97) (1553) _c231, etc.
c231:=922014138575203378678642678703245976645128990649760611223431556044281651224551397420512818296706783874556454242400222424088780517139111803403260652904784384534012886890804276784138532709437357300976962369044109153940982397964463233
Found probable prime factor of 12 digits: 465266047121
Found probable prime factor of
Composite cofactor
22327999684594547306459713871446356290891568171348978419833005355240777166457268798435710862298360946145275352417551980980391889910705469173712623883004107669898621477845591094673 has 179 digits
Soit a=truc(3,64)=(3^64-1)^63+1= (29) (43) (127) _C174 (19531),(3)
(7) _c42 (239803687) (9044179) (2287), (7) _c356 (11632783)
(1723), (3) (19) _C177 (72577)
C174:=529867348090099868146406724786351613699468339767960964877963117507835024337137548647995798851412388611129978239910675066590444837466749242056429867052960253334739754982225539
Found probable prime factor of 8 digits: 21092639
Found probable prime factor of 22 digits: 2732654449408112857139
Found probable prime factor of 28 digits: 5134512751605648050975129789
Found probable prime factor of
Probable prime cofactor
15712972048094993732630324458766139720001345137706786655534580541 has
65 digits
Soit a=truc(15,100)=(15^100-1)^99+1=(23) _c1175, (3) _c229 (2224171),
(331) (991) _c2339 (102871957), (3) (19) (127) _c695 (12987577)
c229:=2477207479083529743825156419411562198342080844294359527032813034352442105085530333254585129268218428528849591971219282951652039046088208376211258576159293895675299612718384435197242837235232431484447012334811619390514096879980481
Found probable prime factor of 29 digits:
93893927988120634518461559757
Found probable prime factor of 44 digits: 70149667684408457592487504451135711760318733
Composite cofactor 376096469547244061307263497569618339882905826703580805196802295618244452479268461369150990468955274126179083017547255747004579641652117744098153555269704601 has 156 digits
