Problems & Puzzles: Puzzles

Puzzle 1195 Follow-up to Puzzle 1194 Related to the challenge in Puzzle 1194, now, let's increase the difficulty grade: Q. Find the a, b & c values, for the smallest three primes with double 666, either contiguous (666666) or isolated (...666...666)? From puzzle 1194 I can see that only one puzzler was aware that he already has gotten one of the now asked solutions

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From2 to 8 Nov, 2024, contributions came from Michael Branicky, Gennady Gusev, JM Rebert, Simon Cavegnm Emmanuek Vantieghem

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Michael wrote:

The a, b, c values for the smallest primes (also shown, each 203 digits) are:

 

( 9, 11, 101) 27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902
120368830900788923839991099564447458450075226030128555294655577015766113909738825769262480452416194899601201
 

( 2, 22, 101) 273186196771574135419986665791560614201471776660881280465910305960827252944980667223385057449021203688309007
88923839991099564447458450075226030128555294655577015766113909739167197139844671973305847233689

(14, 22, 101)27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902120368830900788923839991
099564447458450075226030128555294655577015766113909739167197139844683085312672791701

 

And the (a,b,c) values for the three smallest primes containing 666666 are (with 1210, 1235, and 1385 digits, respectively):

( 77, 151, 455) 24841938311260852622256472601221433950974992555273694535905227066920332021837834312069844909012068384314286
73521757924271191909561215657024636389788364901299607448592204916484260343256309111390705926658071699606009859852938654815
64181231787150274575077194618354573616404121129324735348504502466604154484733963451263511687458240852910329576901488178463
51133009692279544329772221272844135586450093808235218561311592805749978284630956059018749965622293312337910581319157964181
77667192755151350392820641807879282902336668159285761916576962987908815599950407259775231381887213057094010069245691289334
06244864911200246895176985831337090450322856816338479223073261133092000312434168477809499061950224826397091682788706453525
89375495064482765145847187558037938055629347993660624393655855580775975767578295165102080989222017319358281653118702075602
1930435351788433358345353071438307434083528072127278744124393042377119618089584004386267984106719003935118803697980564397
9820972352008583067842354459907464372516720711007154871136410498353408937895631541268291423599694708125609843397266666685
3808451133759248816445420531515454028930769917494113698546115037516668925510721226294615623986838116748753716849681669167
54191323

(321, 415, 463) 145881036819761568454808810233604435618378330352396693882215835085196155121415474676263762278059884021705
0125465918332556955444593318194341864305940446066769431598605695319841803353883234448883087943048783858687591717019681269
8853695111037247638618711324835121306353604324982007882751205329123005518387217521376528011887718245688957330011133685222
6208190219171722174291273178094900422305363682358548043402211002375998757399726730575729696271131605750678458570399661776
3778487052376273161830230444093435947450967299482817821901194429809747193635375662288955316401604454355671441841229897517
5320221440922767882970423773366167173216924953330543908000292938613058484307039995111147543484772818234581842881521108547
6202945914331421636070604010493984966412997992661842825194942364592774958381326767122415796936528941313434321270536515280
0894287498319270555873031947697965768771815227320379177826292913642571718819108612048055111218671974419642408152558054741
9533773595110607464969404684461341424211807332399300456259822751845651140496139887510666666259493963254914520900545311537
7307674394233505068591019024272385201973577529643328016263152476341596718818315078311018482659229664791314043851202991883
94429895835023472156348905603766975329743

 

(193, 455, 511) 100711709734120341230618316897034614605007788541912943950455628197911091935732779524401044119099975030824
3643135548997646812486776143812286506822158715340413506865743147310526103685810529444030239589379418619003704043684574686
3477083599441664306796794561631100979080043697594461786290458297558497315425559554750666568758253103532619949225496885167
6279371693768354224963464607535456159716421043127798054470016646029218094022985327871690370406327267742793337397476411136
3248352822502486666669697141456174055151215497047807866143214083079051145316442484880502317288232841298010212786452129474
5315647660569052477285588843396067574919556984599314170493589764403921380054891854569444525824501487243057184696776191135
0365373930428760411673693477849036766782043447303725155742811281756880288258048821318101313720304236460672583486544144689
3124707902139083094067669301283800172635985949851016056437858653074917156146070160775980554801147081266082433528871390285
7522295711314577083086053059001814981109779461530382108301850486658858111911182191805604749632716195094123085252909087422
6541748130904411383754944422246178578365472961265337899286953972761060962832970159809346937197641440475083252808107704567
9348639670824197642227917838627328520527111285635596461277593057400147877061818531488774973247407964321856976161031232322
7799286841620250815022677136789486457461075175686906987614789302322679

 

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Gennady wrote:

The 3 smallest prime sums of a^a+b^b+c^c are:

a, b, c -> sum

101, 11,  9 -> 2731861967715741354199866657915606142014717766608812804659103059608272529449806672233850574490212
0368830900788923839991099564447458450075226030128555294655577015766113909738825769262480452416194899601201

 

101, 22,  2 -> 27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744902120368830900
788923839991099564447458450075226030128555294655577015766113909739167197139844671973305847233689

 

101, 22, 14 -> 273186196771574135419986665791560614201471776660881280465910305960827252944980667223385057449021203688309007
88923839991099564447458450075226030128555294655577015766113909739167197139844683085312672791701

 

The 3 smallest primes a, b, c giving prime sums are:

101, 47,  7 -> 2731861967715741354199866657915606142014717766608812804659103059608272529449806672233850574490212036883090078
8923839991099568325382713539674652795203480809907770664458811083031686904806080302405586396507

101, 61, 37 -> 27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744910157849393446
732697903952737999705597903768609021162422099991002228632581331395451798779975896520227960384279

101, 73, 67 -> 27318619677157413541998666579156061420147177666088128046591030596093258699644874231395899646723199242185886
706719806463298037883635576598177462597838072741789864241966418273053323323586931529749314801657

 

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Rebert wrote:

I found:

Occurences,  [a,b,c],  digits,  positions of 666,  p

1 [1, 5, 7] 6 [3]

826669

 

2 [9, 11, 101] 203 [24, 45] 

27318619677157413541998666579156061420147177666088128046591030596082725294498066722338505744
90212036883090078892383999109956444745845007522603012855529465557701576611390973882576926248
0452416194899601201

 

3 [27, 65, 101] 203 [24, 45, 168] 

2731861967715741354199866657915606142014717766608812804659103059608272529449806672234541399
7066881289682306327618308277181794826182709529412319245853024642706120668434666263155084719
419499566212188293529

 

4 [110, 131, 148] 322 [17, 125, 173, 255] 

1580279398659915666120428225194503764499868600846167057890676901404618074529511956424458899
8390816144765450698170935563619156666618970245535344475895328243793190556629890086669345907
778274175097042622741165783999

36396787481351769151262771910304807063345066687598385335670331079805997341312974258253823239
646700501197810587

 

5 [80, 110, 383] 990 [459, 643, 662, 879, 906] 

23180907286633588931573385772681604644233546890225817407109322472164308563120016809202797609
91522760557655386222338870423233266140789337362964278620346993099208001833924701035171640061
59796565597489193551472596452442269271776215165977237050521019779720665291739658102892562081
27660737601588396609725288964889186635779711170012080772471082696631808631395143573546020685
876169013292247083658094538306054909978091258265500071039982670587470618601685656176932568666
287684454642960287062699728922908238671410592899337589228219249665580496652281628322638161489
174088085682146822363550063204990231872068439176104769640437346754608322826204464242442166668
758651162122906668438105584073934454082459495507929985688654982990934703757803456496018675290
776459013187157452072160244456739575153449932454142646320388863698045217397024151378360278461
258572571028272207412933679831283064316332594666272398151319611013847423666193059838012223472
0830756645877500783754621910522201767096476449713048518180994687

 

6 [51, 158, 314] 785 [161, 194, 237, 486, 638, 688]

108619924100838391584085095970680902715034237125900324003457605520705413920904590056907090922
501817711859727614940664656171603016602835011561067121677502082366966637098753405856330559210
480927066673953117546614993183750709642593808450036662668698967836711479386925244459926497192
627094024187760914099518544631858673338789645579343247657613898015389313320964627500971455713
929976323393885444737332824123875973886744102307789237274720476980133215387431583115046401512
490644069947491642336660470171081201268104441551845202444382257785012434164870489076149107017
680450369065709739971275913833301775810201994710715394897993153459129013970988166681325712387
674577476085381534713348624388778000666544382040927327544866853083121426483601367072030550753
42550239619886564632570655698493229254251

 

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Simon wrote:

Smallest prime containing '666666':
77^77 + 151^151 + 455^455

Smallest prime containing '666' twice:
9^9 + 11^11 + 101^101

Additional Findings:
Smallest prime containing '666' 16 times:
935^935 + 1886^1886 + 1888^1888

Smallest prime containing '666666666' or '6666666666':
256^256 + 1360^1360 + 1865^1865

With more terms, smaller primes containing '666666' can be found e.g:
3^3 + 29^29 + 32^32 + 55^55 + 58^58

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Emmanuel wrote:

The smallest three solutions (with 2 separated blocks '666' inside) I found are obtained for  (a,b,c) = (9,11,101), (2,22,101)  and  (14,22,101).
The smallest three solutions (with a block of ' 666666' inside) are obtained for  (a,b,c) = {77,151,455}, {321,415,463}  and  {22,28,539}.

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