Website for factorization of Phi(n,2)

sweety439 · 2022-02-05, 14:27

Like https://stdkmd.net/nrr/repunit/phin10.htm, but for Phi(n,2) instead of Phi(n,10)

n range: <= 2^24 (https://stdkmd.net/nrr/repunit/phin10.htm is <= 300000)
number of n values per page (L & M counted as one n value): 256 (https://stdkmd.net/nrr/repunit/phin10.htm is 100)
number of pages: 65536 (https://stdkmd.net/nrr/repunit/phin10.htm is 3000)
shown pages 1, n-16 to n+16 (in page 1 only shown pages 1~17, not 1~33), 65536 in page n (https://stdkmd.net/nrr/repunit/phin10.htm is shown pages 1, n-10 to n+10 (1~21 in page 1, etc., must shown 21 pages in a range), 3000 in page n)

sweety439 · 2022-02-05, 14:44

list for n<=1280

Fermat number: n is of the form 2^r, Phi(n,2) = 2^(2^(r-1))+1
Mersenne number: n is prime, Phi(n,2) = 2^n-1
Wagstaff number: n is twice an odd prime, Phi(n,2) = (2^(n/2)+1)/3
Saouter number: n is of the form 3^r, Phi(n,2) = 4^(3^(r-1))+2^(3^(r-1))+1
nega-Saouter number: n is of the form 2*3^r, Phi(n,2)/3 = (4^(3^(r-1))-2^(3^(r-1))+1)/3 (for these n, gcd(Phi(n,2),n) is 3)
Gaussian-Mersenne number: n is 4 times an odd prime, Phi(nL,2) = 2^(n/4)-2^((n+4)/8)+1 or (2^(n/4)-2^((n+4)/8)+1)/5, Phi(nM,2) = 2^(n/4)+2^((n+4)/8)+1 or (2^(n/4)+2^((n+4)/8)+1)/5 (for some of these n, gcd(Phi(n,2),n) is 5)

? of 18874368 Phi(n (L/M if n == 4 mod 8),2)/gcd(Phi(n (L/M if n == 4 mod 8),2),n) factorizations were finished.
? of 18874368 Phi(n (L/M if n == 4 mod 8),2)/gcd(Phi(n (L/M if n == 4 mod 8),2),n) factorizations were cracked.
11 of 23 Fermat number factorizations were finished.
22 of 23 Fermat number factorizations were cracked.
? of 1077871 Mersenne number factorizations were finished.
? of 1077871 Mersenne number factorizations were cracked.
? of 564162 Wagstaff number factorizations were finished.
? of 564162 Wagstaff number factorizations were cracked.
6 of 15 Saouter number factorizations were finished.
15 of 15 Saouter number factorizations were cracked.
6 of 14 nega-Saouter number factorizations were finished.
11 of 14 nega-Saouter number factorizations were cracked.
? of 295946 Gaussian-Mersenne number factorizations were finished.
? of 295946 Gaussian-Mersenne number factorizations were cracked.
? (probable) prime factors were discovered.
? composite factors are remaining.
? factors are unidentified.

(Note: "Cn" in the list means composite number with n binary digits, not decimal digits)

First composite factor:

n=1207 (C1120)
n=1213 (C986)
n=1217 (C822)
n=1229 (C943)
n=1231 (C1090)
n=1237 (C1007)
n=1243 (C1120)
n=1249 (C1083)
n=1253 (C889)
n=1255 (C731)
n=1259 (C1027)
n=1265 (C740)
n=1277 (C1277)
n=1283 (C1152)
n=1291 (C1156)

Smallest composite factor:

n=3030 (C576)
n=3390 (C597)
n=3066 (C619)
n=3120 (C626)
n=1785 (C627)
n=1635 (C655)
n=6132L (C661)
n=3198 (C669)
n=5644M (C670)
n=1995 (C674)
n=5388M (C675)
n=5388L (C677)
n=3312 (C680)
n=1701 (C681)

First blank Phi(n,2):

n=1207 (C1120)
n=1243 (C1120)
n=1277 (C1277)
n=1387 (C1296)
n=1537 (C1456)

Smallest blank Phi(n,2):

First blank Fermat number:

n=2097152 (C1048577)
(next n is 33554432, which is beyond the table range)

First blank Mersenne number:

n=1277 (C1277)
n=1619 (C1619)
n=1753 (C1753)
n=2267 (C2267)
n=2273 (C2273)
n=2423 (C2423)
n=2521 (C2521)
n=2713 (C2713)
n=2719 (C2719)
n=2851 (C2851)

First blank Wagstaff number:

n=2246 (C1122)
n=2518 (C1258)
n=2866 (C1432)
n=3394 (C1696)
n=3554 (C1776)
n=3826 (C1912)
n=3986 (C1992)
n=4226 (C2112)
n=5582 (C2790)
n=5602 (C2800)
n=5774 (C2886)
n=5818 (C2908)

First blank Saouter number:

(first n is 43046721, which is beyond the table range)

First blank nega-Saouter number:

n=354294 (C118097)
n=1062882 (C354293)
n=3188646 (C1062881)

First blank Gaussian-Mersenne number:

n=4852M (C1214)
n=5204M (C1302)
n=5884L (C1471)
n=6196L (C1547)
n=6452L (C1611)
n=6628M (C1655)
n=6772L (C1691)
n=6796M (C1700)
n=7036L (C1759)
n=7132M (C1781)
n=7484L (C1871)
n=7492M (C1871)
n=7628M (C1908)
n=7804M (C1949)
n=7916M (C1980)
n=7996L (C1999)

sweety439 · 2022-02-05, 15:52

Related links:

https://www.mersenne.org/ (the Mersenne numbers 2^p-1 with prime p)
http://www.prothsearch.com/fermat.html (the Fermat numbers 2^(2^n)+1)
https://www.alpertron.com.ar/MODFERM.HTM (the Saouter numbers 4^(3^n)+2^(3^n)+1 and the nega-Saouter numbers (4^(3^n)-2^(3^n)+1)/3
http://web.archive.org/web/201606030...os/fermat6.htm (Mersenne-Fermat numbers Phi(p^r,2) with prime p)
https://homes.cerias.purdue.edu/~ssw/cun/index.html (The Cunningham Project, n <= 1500, even n <= 3000, n == 4 mod 8 (L,M) <= 6000)

sweety439 · 2022-02-05, 15:57

If and only if Phi(n,2)/gcd(Phi(n,2),n) is prime power (it is conjectured that it cannot be a true power (i.e. >=2 power) of a prime, i.e. unique prime base 2 cannot be Wieferich prime), then this prime is unique prime base 2, such n are listed in A161508, the known examples with gcd(Phi(n,2),n) >= 1 (if so, then the gcd must be prime, in fact, the gcd must be the largest prime factor of n) are {18, 20, 21, 54, 147, 342, 602, 889, 258121}, see A333973 (they are subsequence of A093106, i.e. of the form znorder(Mod(2,p))*p^r for some r>=1 and odd prime p: 2*3^2, 4*5^1, 3*7^1, 2*3^3, 3*7^2, 18*19^1, 14*43^1, 127*7^1, 359*719^1, note that for all these primes p except 43 and 719, the Mersenne number 2^p-1 = Phi(p,2) is prime, and for all these primes p except 719, the Wagstaff number (2^p+1)/3 = Phi(2*p,2) is prime)

All Fermat primes (only 5 known: 2^(2^n)+1 for n = 0, 1, 2, 3, 4), Mersenne primes, Wagstaff primes, Saouter primes (only 3 known: 4^(3^n)+2^(3^n)+1 for n = 0, 1, 2), Mersenne-Fermat primes (only 8 non-Mersenne ones are known: Phi(n,2) with n = 4, 8, 9, 16, 27, 32, 49, 3481) and related primes (4^(3^n)-2^(3^n)+1)/3 (only 2 known: n = 1 and 2, note that n = 0 gives 1 instead of a prime) are unique primes in base 2, but it is not known whether there are infinitely many unique primes in base 2 (although this is very likely, since Mersenne primes are excepted to exist infinitely many), however it is known that Phi(n,2)/gcd(Phi(n,2),n) is composite for all n > 20 which is == 4 mod 8, this is because the Aurifeuillian factorization, for n == 4 mod 8, there are Phi(nL,2) and Phi(nM,2), and they are in fact related to the Gaussian-Mersenne numbers.

sweety439 · 2022-02-05, 16:10

If and only if Phi(n,2)/gcd(Phi(n,2),n) is divisible by two primes, then they are "bi-unique primes" in base 2

One of the two primes is n+1 for these n:

{28, 36, 58, 60, 66, 82, 106, 138, 1290, 3010, 6658, 9546, ...}

One of the two primes is 2*n+1 for these n:

{11, 23, 35, 39, 48, 83, 96, 131, 231, 303, 375, 384, 519, 771, 848, 1400, 1983, 2280, 2640, 2715, 3359, 6144, 7736, 7911, 11079, 13224, ...} (two large known examples are 130439 and 406583, see http://www.primenumbers.net/prptop/s...&action=Search)

One of the two primes is 3*n+1 for these n:

{36, 94, 214, 1086, 5894, ...}

One of the two primes is 4*n+1 for these n:

{28, 70, 88, 144, 1470, 7830, ...}

One of the two primes is 6*n+1 for these n:

{37, 72, 121, 153, 221, 245, 688, 6057, ...}

(only for n+1 and 2*n+1 the sequences are excepted to be infinite, for 3*n+1, 4*n+1, etc. the sequences are excepted to be finite)

The bi-unique primes with at least one prime <= 23311 are

(p,q,period length)

Code:

23,89,11
29,113,28
37,109,36
47,178481,23
59,3033169,58
61,1321,60
67,20857,66
71,122921,35
79,121369,39
83,8831418697,82
89,23,11
97,673,48
107,28059810762433,106
109,37,36
113,29,28
139,168749965921,138
167,57912614113275649087721,83
193,22253377,96
223,616318177,37
251,4051,50
263,10350794431055162386718619237468234569,131
281,86171,70
283,165768537521,94
353,2931542417,88
397,2113,44
433,38737,72
463,4982397651178256151338302204762057,231
571,160465489,114
577,487824887233,144
601,1801,25
607,1512768222413735255864403005264105839324374778520631853993,303
631,23311,45
641,6700417,64
643,84115747449047881488635567801,214
673,97,48
727,1786393878363164227858270210279,121
751,2139731020464054092520609592459940706818275139793055476751,375
769,442499826945303593556473164314770689,384
919,75582488424179347083438319,153
1039,19709014643115560219397264671577125505264032974428376489237001990435774189483906244488746953221813209,519
1291,83861817871925183739792206470703862766563053456867813459969184678546547694793573468589875745315081,1290
1321,61,60
1327,2365454398418399772605086209214363458552839866247069233,221
1429,14449,84
1471,252359902034571016856214298851708529738525821631,245
1543,4965395030068548134274243124972075225434447114375481299036593442726326832727934403424309955102162841656341524725641213163998408700663382552888660520657,771
1697,99335205800663868215396640964567095667094665346141013294320587365443384719802857319737050495099341955640963272958071602273,848
1753,1795918038741070627,146
1777,25781083,74
1801,601,25
2113,397,44
2281,3011347479614249131,190
2801,1114513219367157067542813609361306957257890531134775327875067038594481393220804051366788787128409731513666376851495151281817670381468528387601,1400
2971,48912491,110
3011,631215008947706187342830494125660733360092019659681922883823392015121754384870744044074337887482936870852519582960673945561810148710850934449712549090934572292098088972061029650939105592263256293676274598529593937386833315889748213948490958132757432166701901197169972066727635929332437543971934775961,3010
3259,960843850986532976532466235773483492840618819232206145010143480044702708779967241439519037158800917230289,1086
3361,88959882481,168
3967,3296810823331827444014404831943558588631803435050404237042485765714486337505843011741487225539321479275976317423474114853376321380782906502106758766783934866952124117240484839332668914566806988602931402117416523955329423560856334826333176954575294550104263404414368761262079586842542586869780254842277261781328657636993064897732127711363870426953852536828242291991249685206783121190349820804553,1983
4051,251,50
4129,33770734168253651800370989375796994825389296318018601048482005531172856260013942500368975908606689,688
4177,9857737155463,87
4523,106788290443848295284382097033,266
4561,51049903050598156013062477654241640657829025002976204451060261008689478158715729745160924860467530309657376827104233308157772350164622158651187694109112727796663977157921,2280
4871,82033219963138371097689272308258116841679442057301643873942124991182012434598644913857356023840478815121709542915222280972560231358838127531337,487
5153,54410972897,112
5281,860573414369008969457638101533827364704684164286188824383996871471626764468219429432850649798234488791036977295952185305281368586922360240161830570497885830241813162641447900350702795124321,2640
5347,242099935645987,198
5431,9496792988973395279834809661569251320544014305629535339041176003993804676485199470146061766714674245857484692035146769896735263094609746643655454990307740324793870789211183639302657602006206727998357155351588374904926882678074633252877772442134938712719216422825919305415646969838444956229559805502059906228691551284152908049964563516405524127028294884716508273187514617905113012642316992618568629664046915514871575875088457784721,2715
5881,23618256244840618857212522155851714598259422753496906641681177748710460515038403366198473773770441,1470
6043,4475130366518102084427698737,318
6659,67348091890626757137914773048080151982788009808953349522971703676209072447919253736713943719839321930921001920443146832964494535806286153336758808764827052922284527987735369082658226155752758837734585788583920437031679967832798697874531506375848295306002069974292647012532975382657869395896549536084026643086849168707638035984652951756449232381922841508279043449553683642165895518852273616853752192626547082225232322662203349617421624450106130361133033050996977517456780759336980017504035553443204836636663583950658161718264375035034007950563418777684540446570742277682688819816930529249402141227674467861784042184664703273065772145626307008333021727910295689307897812592176340797189662547498298629041419687123412984210580325376481846316396566413701109848755348878790562296275295266190788880122151835567771665815565611863261470167857288685014242115505182159685753576612239477286620238583071292970734389580521730540789853959607322402465845627773409459421340250476125665859926003121138412497735360569,6658
6719,215006106257113223254503015023149432126193150293791416185445173578281597218315377296589584591228602041183907532584815068471747291177386898925622477208530115714962355294842135137890474394949339249259335407710018584480055157825387089416912233252714054247018216597994795059161567922302450277281351135838393171424038832688432240078361264161523904355539085927738753968157018550258476163852090826756157915705283413226000816151712543838581066281600650278690534719371112997393190721068136840596790525950480851370510277560248341182341805553054000587378384785994695875587394905226703149605689830257768229987714770949192483302583569799141079867597051190134078718011730508482542567284418838119000563443985593691221203060137047648713095502877775290062907208508727269017130916691676838817452529964938878349395785642430571852241837461604136374448443730175081889502056290512497717177577492736555784081731998565765598518104822516520340301701034123926767472784665776779480628821628279687651736198541330802238405154786248043073,3359
7487,26828803997912886929710867041891989490486893845712448833,197
8929,197107422273014301919781414466039325387889623676342705850752210599969,496
8969,10508537584872980049787749414505440238543661684506416445249892188329191267897669657242625405655025902294996965713681247700894953567276596965114308183649957469931262029470372188492494505614207827774171575432114297123003373257035070542940532411186322417809411123684246738342720455933424175399671044286557638075591,1121
9547,1621441292160739312484402643488810210953460916758334047593952342310982348899125523375207637304333778211869062392988099059802528019593682234941755422758885656395068722385037980657466257618112582188770921312100125511337836412531718154395821529210922389443733616354268820219577863577759459082447218927273695668223251258943006743614909639761127161704816862626236353032622115795192245125083091261029053988053316433377173895018793740052548266015018756763150731725385456332232982433576015547722563978072554378000015707071821371450842910648052930276764535303424167478747579771592484270800978561959411183367133498969236434846108865206764889977963554070295936092795484663326925724277620386077381551473009733178990983013487085676185144378484902884955972104873606558173086267499547566081780818064857671567480196001693236835136811036110768546793929610732909274227296407079545788520811837495181586420117807667033593394473,9546
11447,13842607235828485645766393,97
12289,2629750677134917999895424907532748959267971573742003745799523592442400125223198868353584621329881476417016678405232160427546550504556182040449845584380200432958814986353730660895939274993054935861018605125239447333126553718172329413446016302260246600258901932788372124899306362726101864155549873799652566158494146315907080383262433417486017334322921176453095565205010625614231712661970815446134764060685945211556960683423424525993942662827506654887366765611019050113870761761408961221324345522474182549452800034874415507055728422655263711997852718353699251231202833814683318936070569832083361696535153167990509569,6144
13367,164511353,41
14323,70180796165277040349245703851057,462
14449,1429,84
15473,9204491083726541729266567601471444329696631442344981431080754304338517202479825704708168361584443220173159746955658319872226827894696158427067368155867113488065513258508747555444435213793714645707333231354709392097881839581182784326394055584833662570305406399842719309326889563391133205121809640737536847612190163876346404787817387525638094982013601782945250606972114988322647360650287694307006008987227557105771507174425909017893011796996993964961154355753480435864294553880686455000837365767828673806742459951938371437326566489572167787070595907923956337478965698716896738108637875683461439370672939945722664550460118112314840960437168628395852506252366614840117258891067154642808373550316537689707097385866475178034598724990199374366941091632039248444592165926072488503954436734636696960611114179555123828676842815894123511610807438661339141915801655793131755397177593505570978799199492340891265152845166543749678711505995868169443966984963587657337523315541043519335991932587032243221833506118827959949035567629728410565768244083641612636803390113309291344892066167245002709645081207176400738835471592939623580395456780283517466070695133155570570494491777,7736
15823,1031618980932934769383824723184453613314324106752135050673854379301654552714215394291015163128024361973584332732468413818125136605907925760742727247647220415724847868638603310295946886910704468341772642665497565950750086698629103691976713579022960064415192185939073009721021079287407784568315198906783459962238739583130435870717508606997094648266422484691784392564301805192820221466239073651256038959925151306122808986987582876448417134983493001649748925736318938290019558037650702924040573324690113487608873342735473727754886620731294191461735069245133317613667710164108542780148358750860213557679286663895342786365468424325384132870658240860855147599880535151997621542107653002820264151246209136053497024192539522831658986624957767737721911252822787888843752829459086074452500078387660213490866191921204625121426981373655768778637038215172918926689432919684828829098866986523456559402508436794953510192587526206390672734345055987579561696548902294039249528563929351059910627127866642657962175417293301256840237920081083939786571599063209033631856956878768239632309569341130784199531718664271487988377765144869060899804994540481914283199459828428853508194150203945298422277045966978011672616271671679440272231090543293690647223138433446445413262172456138967318288034331777417036012303446606440531466158280575773170897474287947075418886039018360314481465874051574744225084498561391552477465092787861625989825113777108770311727266661950047388314382248455070451134432643654587397359448294711228969814351285436562223899913063645073334532624007490078765078132711445999083340183325487,7911
16729,136364260462350955061807337963242197493167687688479364067250955475008708422731598707787831617704017498828261921599586128766571055323409303455188555673114202934877515988195740967406991575319,697
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19211,55032010963083225277133748781276216891770769029556852777446173842960223493066409672741791064238834027799294487913679490115371458761,1130
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20857,67,66
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23041,14768784307009061644318236958041601,480
23311,631,45

sweety439 · 2022-02-07, 08:23

Like the list of PRP factors and list of proven primes > 10^1000, but for Phi(n,2) instead of Phi(n,10):

Phi(n,2)/gcd(Phi(n,2),n) itself for n up to certain limit:

A072226 (Phi(n,2) is itself prime or PRP, list of all such n <= 100000)
A161508 (Phi(n,2)/gcd(Phi(n,2),n) is itself prime or PRP, list of all such n <= 10000)
(The known n such that Phi(n,2)/gcd(Phi(n,2),n) is prime or PRP but gcd(Phi(n,2),n) > 1 (i.e. n is in A093106) are 18, 20, 21, 54, 147, 342, 602, 889, 258121, see A333973)

Large known special values of n:

A000043 (all known prime n)
2*A000978 (all known n which are twice odd primes)
8*A057182 (all known n which are 8 times odd primes)
16*A127317 (all known n which are 16 times odd primes)
4*A007670L (the Aurifeuillian L-part, all known n which are == +-4 mod 32)
4*A125742L (the Aurifeuillian L-part, all known n which are == +-12 mod 32)
4*A007671M (the Aurifeuillian M-part, all known n which are == +-12 mod 32)
4*A124165M (the Aurifeuillian M-part, all known n which are == +-4 mod 32)
A317299 (all known semiprime n)
(the only known power-of-2 n are 2, 4, 8, 16, 32, see http://www.prothsearch.com/fermat.html)
(the only known power-of-3 n are 3, 9, 27, see https://www.alpertron.com.ar/MODFERM.HTM)
(the only known n which are twice power of 3 are 18 and 54, see https://www.alpertron.com.ar/MODFERM.HTM)
(the only known true (i.e. >= 2nd power) prime powers n are 4, 8, 9, 16, 27, 32, 49, 3481, see A297625) (A245730 lists all prime powers n, including the primes themselves)

Divisors of Phi(n,2)/gcd(Phi(n,2),n):

http://www.primenumbers.net/prptop/s...&action=Search (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is prime)
http://www.primenumbers.net/prptop/s...&action=Search (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is twice an odd prime)
http://www.prothsearch.com/fermat.html (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is power of 2, although in fact no such n are known)
https://www.alpertron.com.ar/MODFERM.HTM (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is power of 3 or twice a power of 3, although in fact no such n are known)
https://web.archive.org/web/20111107.../factoredM.txt (list of all n such that Phi(n,2)/gcd(Phi(n,2),n) is completely factored, from the wayback machine, original link is dead)

sweety439 · 2022-02-10, 08:17

this is the list of the n (for every odd prime p, there is exactly one such n) such that p divides Phi(n,2), this n is the multiplicative order of 2 mod p, i.e. znorder(Mod(2,n)), but if this n is == 4 mod 8, we include the Aurifeuillian L or M

sweety439 · 2022-02-20, 03:26

By Fermat little theorem, for all odd primes p, the multiplicative order of 2 mod p must divide p-1, and this file is

Code:

p,multiplicative order of 2 mod p,ratio of p-1 to multiplicative order of 2 mod p

For the multiplicative order of 2 mod p for odd primes p, see A014664
For the ratio of p-1 to multiplicative order of 2 mod p for odd primes p, see A001917
For the smallest odd prime p such that multiplicative order of 2 mod p is n, see A112927
For the smallest odd prime p such that ratio of p-1 to multiplicative order of 2 mod p is n, see A101208
For the product of the odd primes p such that multiplicative order of 2 mod p is n, see A064078

sweety439 · 2022-03-04, 13:22

Related conjectures:

* 2^Phi(n,2)-1 is prime only for n = 2, 3, 4, 5, 6, 7, 8, 12
* (2^Phi(n,2)+1)/3 is prime only for n = 2, 3, 4, 5, 6, 7, 8, 10, 12, 14

and stronger conjectures:

* Phi(Phi(n,2),2) is prime only for n = 2, 3, 4, 5, 6, 7, 8, 12
* Phi(2*Phi(n,2),2) is prime only for n = 2, 3, 4, 5, 6, 7, 8, 10, 12, 14

see related post https://www.mersenneforum.org/showpo...11&postcount=5

sweety439 · 2022-03-04, 13:37

A generalization of the New Mersenne Conjecture, for any positive odd number n:

* n is of the form 2^k+-1 or 4^k+-3
* Phi(n,2) is prime
* Phi(2*n,2) is prime

The only n such that these three statements are all true are {3, 5, 7, 13, 15, 17, 19, 31, 61, 127}

The only n such that exactly two of these three statements are true are {1, 9, 33, 49, 63, 65, 85, 129, 345} (the New Mersenne Conjecture is that this set contains no primes)

If we consider the unique primes, i.e. using Phi(n,2)/gcd(Phi(n,2),n) instead of Phi(n,2) itself (I also conjectured that Phi(n,2)/gcd(Phi(n,2),n) cannot be true prime power, i.e. p^r with p prime and r>1, in fact, I conjectured that Phi(n,2)/gcd(Phi(n,2),n) cannot be perfect power, the cases that n is prime or n is power of 2 are covered by Catalan's conjecture, which is already be proven), then the conjecture become:

* n is of the form 2^k+-1 or 4^k+-3
* Phi(n,2)/gcd(Phi(n,2),n) is prime
* Phi(2*n,2)/gcd(Phi(2*n,2),n) is prime

The only n such that these three statements are all true are {5, 7, 9, 13, 15, 17, 19, 31, 61, 127}

The only n such that exactly two of these three statements are true are {1, 3, 21, 27, 33, 49, 63, 65, 85, 129, 345}

sweety439 · 2022-04-14, 04:45

this is the primitive prime factors of 2^n-1 and 2^n+1 (equivalently, the prime factors of Phi(n,2) and Phi(2*n,2)), for n <= 2^13

2022-02-05, 14:27	#1
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 E60₁₆ Posts	Website for factorization of Phi(n,2) Like https://stdkmd.net/nrr/repunit/phin10.htm, but for Phi(n,2) instead of Phi(n,10) n range: <= 2^24 (https://stdkmd.net/nrr/repunit/phin10.htm is <= 300000) number of n values per page (L & M counted as one n value): 256 (https://stdkmd.net/nrr/repunit/phin10.htm is 100) number of pages: 65536 (https://stdkmd.net/nrr/repunit/phin10.htm is 3000) shown pages 1, n-16 to n+16 (in page 1 only shown pages 1~17, not 1~33), 65536 in page n (https://stdkmd.net/nrr/repunit/phin10.htm is shown pages 1, n-10 to n+10 (1~21 in page 1, etc., must shown 21 pages in a range), 3000 in page n)

2022-02-05, 14:44	#2
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 E60₁₆ Posts	list for n<=1280 Fermat number: n is of the form 2^r, Phi(n,2) = 2^(2^(r-1))+1 Mersenne number: n is prime, Phi(n,2) = 2^n-1 Wagstaff number: n is twice an odd prime, Phi(n,2) = (2^(n/2)+1)/3 Saouter number: n is of the form 3^r, Phi(n,2) = 4^(3^(r-1))+2^(3^(r-1))+1 nega-Saouter number: n is of the form 23^r, Phi(n,2)/3 = (4^(3^(r-1))-2^(3^(r-1))+1)/3 (for these n, gcd(Phi(n,2),n) is 3) Gaussian-Mersenne number: n is 4 times an odd prime, Phi(nL,2) = 2^(n/4)-2^((n+4)/8)+1 or (2^(n/4)-2^((n+4)/8)+1)/5, Phi(nM,2) = 2^(n/4)+2^((n+4)/8)+1 or (2^(n/4)+2^((n+4)/8)+1)/5 (for some of these n, gcd(Phi(n,2),n) is 5) ? of 18874368 Phi(n (L/M if n == 4 mod 8),2)/gcd(Phi(n (L/M if n == 4 mod 8),2),n) factorizations were finished. ? of 18874368 Phi(n (L/M if n == 4 mod 8),2)/gcd(Phi(n (L/M if n == 4 mod 8),2),n) factorizations were cracked. 11 of 23 Fermat number factorizations were finished. 22 of 23 Fermat number factorizations were cracked. ? of 1077871 Mersenne number factorizations were finished. ? of 1077871 Mersenne number factorizations were cracked. ? of 564162 Wagstaff number factorizations were finished. ? of 564162 Wagstaff number factorizations were cracked. 6 of 15 Saouter number factorizations were finished. 15 of 15 Saouter number factorizations were cracked. 6 of 14 nega-Saouter number factorizations were finished. 11 of 14 nega-Saouter number factorizations were cracked. ? of 295946 Gaussian-Mersenne number factorizations were finished. ? of 295946 Gaussian-Mersenne number factorizations were cracked. ? (probable) prime factors were discovered. ? composite factors are remaining. ? factors are unidentified. (Note: "Cn" in the list means composite number with n binary* digits, not decimal digits) First composite factor: n=1207 (C1120) n=1213 (C986) n=1217 (C822) n=1229 (C943) n=1231 (C1090) n=1237 (C1007) n=1243 (C1120) n=1249 (C1083) n=1253 (C889) n=1255 (C731) n=1259 (C1027) n=1265 (C740) n=1277 (C1277) n=1283 (C1152) n=1291 (C1156) Smallest composite factor: n=3030 (C576) n=3390 (C597) n=3066 (C619) n=3120 (C626) n=1785 (C627) n=1635 (C655) n=6132L (C661) n=3198 (C669) n=5644M (C670) n=1995 (C674) n=5388M (C675) n=5388L (C677) n=3312 (C680) n=1701 (C681) First blank Phi(n,2): n=1207 (C1120) n=1243 (C1120) n=1277 (C1277) n=1387 (C1296) n=1537 (C1456) Smallest blank Phi(n,2): First blank Fermat number: n=2097152 (C1048577) (next n is 33554432, which is beyond the table range) First blank Mersenne number: n=1277 (C1277) n=1619 (C1619) n=1753 (C1753) n=2267 (C2267) n=2273 (C2273) n=2423 (C2423) n=2521 (C2521) n=2713 (C2713) n=2719 (C2719) n=2851 (C2851) First blank Wagstaff number: n=2246 (C1122) n=2518 (C1258) n=2866 (C1432) n=3394 (C1696) n=3554 (C1776) n=3826 (C1912) n=3986 (C1992) n=4226 (C2112) n=5582 (C2790) n=5602 (C2800) n=5774 (C2886) n=5818 (C2908) First blank Saouter number: (first n is 43046721, which is beyond the table range) First blank nega-Saouter number: n=354294 (C118097) n=1062882 (C354293) n=3188646 (C1062881) First blank Gaussian-Mersenne number: n=4852M (C1214) n=5204M (C1302) n=5884L (C1471) n=6196L (C1547) n=6452L (C1611) n=6628M (C1655) n=6772L (C1691) n=6796M (C1700) n=7036L (C1759) n=7132M (C1781) n=7484L (C1871) n=7492M (C1871) n=7628M (C1908) n=7804M (C1949) n=7916M (C1980) n=7996L (C1999) Last fiddled with by sweety439 on 2022-03-04 at 13:11

2022-02-05, 15:52	#3
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵·5·23 Posts	Related links: https://www.mersenne.org/ (the Mersenne numbers 2^p-1 with prime p) http://www.prothsearch.com/fermat.html (the Fermat numbers 2^(2^n)+1) https://www.alpertron.com.ar/MODFERM.HTM (the Saouter numbers 4^(3^n)+2^(3^n)+1 and the nega-Saouter numbers (4^(3^n)-2^(3^n)+1)/3 http://web.archive.org/web/201606030...os/fermat6.htm (Mersenne-Fermat numbers Phi(p^r,2) with prime p) https://homes.cerias.purdue.edu/~ssw/cun/index.html (The Cunningham Project, n <= 1500, even n <= 3000, n == 4 mod 8 (L,M) <= 6000) Last fiddled with by sweety439 on 2022-02-10 at 08:01

2022-02-05, 15:57	#4
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵·5·23 Posts	If and only if Phi(n,2)/gcd(Phi(n,2),n) is prime power (it is conjectured that it cannot be a true power (i.e. >=2 power) of a prime, i.e. unique prime base 2 cannot be Wieferich prime), then this prime is unique prime base 2, such n are listed in A161508, the known examples with gcd(Phi(n,2),n) >= 1 (if so, then the gcd must be prime, in fact, the gcd must be the largest prime factor of n) are {18, 20, 21, 54, 147, 342, 602, 889, 258121}, see A333973 (they are subsequence of A093106, i.e. of the form znorder(Mod(2,p))p^r for some r>=1 and odd prime p: 23^2, 45^1, 37^1, 23^3, 37^2, 1819^1, 1443^1, 1277^1, 359719^1, note that for all these primes p except 43 and 719, the Mersenne number 2^p-1 = Phi(p,2) is prime, and for all these primes p except 719, the Wagstaff number (2^p+1)/3 = Phi(2p,2) is prime) All Fermat primes (only 5 known: 2^(2^n)+1 for n = 0, 1, 2, 3, 4), Mersenne primes, Wagstaff primes, Saouter primes (only 3 known: 4^(3^n)+2^(3^n)+1 for n = 0, 1, 2), Mersenne-Fermat primes (only 8 non-Mersenne ones are known: Phi(n,2) with n = 4, 8, 9, 16, 27, 32, 49, 3481) and related primes (4^(3^n)-2^(3^n)+1)/3 (only 2 known: n = 1 and 2, note that n = 0 gives 1 instead of a prime) are unique primes in base 2, but it is not known whether there are infinitely many unique primes in base 2 (although this is very likely, since Mersenne primes are excepted to exist infinitely many), however it is known that Phi(n,2)/gcd(Phi(n,2),n) is composite for all n > 20 which is == 4 mod 8, this is because the Aurifeuillian factorization, for n == 4 mod 8, there are Phi(nL,2) and Phi(nM,2), and they are in fact related to the Gaussian-Mersenne numbers. Last fiddled with by sweety439 on 2022-03-04 at 13:16*

2022-02-07, 08:23	#6
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵·5·23 Posts	Like the list of PRP factors and list of proven primes > 10^1000, but for Phi(n,2) instead of Phi(n,10): Phi(n,2)/gcd(Phi(n,2),n) itself for n up to certain limit: A072226 (Phi(n,2) is itself prime or PRP, list of all such n <= 100000) A161508 (Phi(n,2)/gcd(Phi(n,2),n) is itself prime or PRP, list of all such n <= 10000) (The known n such that Phi(n,2)/gcd(Phi(n,2),n) is prime or PRP but gcd(Phi(n,2),n) > 1 (i.e. n is in A093106) are 18, 20, 21, 54, 147, 342, 602, 889, 258121, see A333973) Large known special values of n: A000043 (all known prime n) 2A000978 (all known n which are twice odd primes) 8A057182 (all known n which are 8 times odd primes) 16A127317 (all known n which are 16 times odd primes) 4A007670L (the Aurifeuillian L-part, all known n which are == +-4 mod 32) 4A125742L (the Aurifeuillian L-part, all known n which are == +-12 mod 32) 4A007671M (the Aurifeuillian M-part, all known n which are == +-12 mod 32) 4A124165M (the Aurifeuillian M-part, all known n which are == +-4 mod 32) A317299 (all known semiprime n) (the only known power-of-2 n are 2, 4, 8, 16, 32, see http://www.prothsearch.com/fermat.html) (the only known power-of-3 n are 3, 9, 27, see https://www.alpertron.com.ar/MODFERM.HTM) (the only known n which are twice power of 3 are 18 and 54, see https://www.alpertron.com.ar/MODFERM.HTM) (the only known true (i.e. >= 2nd power) prime powers n are 4, 8, 9, 16, 27, 32, 49, 3481, see A297625) (A245730 lists all prime powers n, including the primes themselves) Divisors of Phi(n,2)/gcd(Phi(n,2),n): http://www.primenumbers.net/prptop/s...&action=Search (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is prime) http://www.primenumbers.net/prptop/s...&action=Search (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is twice an odd prime) http://www.prothsearch.com/fermat.html (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is power of 2, although in fact no such n are known) https://www.alpertron.com.ar/MODFERM.HTM (PRP divisors of Phi(n,2)/gcd(Phi(n,2),n) when n is power of 3 or twice a power of 3, although in fact no such n are known) https://web.archive.org/web/20111107.../factoredM.txt (list of all n such that Phi(n,2)/gcd(Phi(n,2),n) is completely factored, from the wayback machine, original link is dead) Last fiddled with by sweety439 on 2022-03-04 at 13:13*

2022-03-04, 13:22	#9
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵×5×23 Posts	Related conjectures: * 2^Phi(n,2)-1 is prime only for n = 2, 3, 4, 5, 6, 7, 8, 12 * (2^Phi(n,2)+1)/3 is prime only for n = 2, 3, 4, 5, 6, 7, 8, 10, 12, 14 and stronger conjectures: * Phi(Phi(n,2),2) is prime only for n = 2, 3, 4, 5, 6, 7, 8, 12 * Phi(2*Phi(n,2),2) is prime only for n = 2, 3, 4, 5, 6, 7, 8, 10, 12, 14 see related post https://www.mersenneforum.org/showpo...11&postcount=5

2022-03-04, 13:37	#10
sweety439 "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2⁵·5·23 Posts	A generalization of the New Mersenne Conjecture, for any positive odd number n: * n is of the form 2^k+-1 or 4^k+-3 * Phi(n,2) is prime * Phi(2n,2) is prime The only n such that these three statements are all true are {3, 5, 7, 13, 15, 17, 19, 31, 61, 127} The only n such that exactly two of these three statements are true are {1, 9, 33, 49, 63, 65, 85, 129, 345} (the New Mersenne Conjecture is that this set contains no primes) If we consider the unique primes, i.e. using Phi(n,2)/gcd(Phi(n,2),n) instead of Phi(n,2) itself (I also conjectured that Phi(n,2)/gcd(Phi(n,2),n) cannot be true prime power, i.e. p^r with p prime and r>1, in fact, I conjectured that Phi(n,2)/gcd(Phi(n,2),n) cannot be perfect power, the cases that n is prime or n is power of 2 are covered by Catalan's conjecture, which is already be proven), then the conjecture become: n is of the form 2^k+-1 or 4^k+-3 * Phi(n,2)/gcd(Phi(n,2),n) is prime * Phi(2n,2)/gcd(Phi(2n,2),n) is prime The only n such that these three statements are all true are {5, 7, 9, 13, 15, 17, 19, 31, 61, 127} The only n such that exactly two of these three statements are true are {1, 3, 21, 27, 33, 49, 63, 65, 85, 129, 345}

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