Puzzle 1194 826669

From Oct 26 to Nov 2, 2024, contributions came from Michael Branicky, JM Rebert, Emmanuel Vantieghem, Gennady Gusev

It is quite easy to find such numbers. For example, searching all 1 <= a <= b <= c <= 100, we find

1^1 + 5^5 + 7^7 = 826669
1^1 + 11^11 + 77^77 = 18188037387806198379277339915556929647807403283187048631478337739929618787870634227045716719924575689062274471430368865388203540672666327842667409
2^2 + 8^8 + 77^77 = 18188037387806198379277339915556929647807403283187048631478337739929618787870634227045716719924575689062274471430368865388203540672666042547774017
2^2 + 16^16 + 27^27 = 443426488243037769966696374692859444423
2^2 + 16^16 + 47^47 = 3877924263464448622666648186154330754898344901344205917642344074630570094614483
3^3 + 8^8 + 22^22 = 341427877364219557396663500827
3^3 + 14^14 + 68^68 = 40794917954274783314474389422963594412010553412954188046665939634971631296545460720786532465498226465248060578657593919291419
3^3 + 40^40 + 56^56 = 79164324866862966607842406018063266761180441458938437285122402918484170424104310169552592050323483
4^4 + 11^11 + 98^98 = 1380878341261486750656911803252309726876604105686729638072729543243701479670593033211008001443536626310535980077544691196522513327846303307992442770355560270350429006522588433404602388277402966611
4^4 + 15^15 + 70^70 = 1435036016098684342856030763566710717400773837392460666392490000000000000000000000000000000000000000000000000000437893890380859631
6^6 + 16^16 + 97^97 = 5210245939718361468048211048414496022534389576033913164940029913016568215580398296261072019231723279851007241838011659882766685337218633992220688288491655299087016195985205236794455652195343009
7^7 + 78^78 + 82^82 = 8565168194859488946144470110697602886084692943398947929638977229295206040649131939371020936733638727783441156639866312959414932746957002009371580398922666231
8^8 + 52^52 + 83^83 = 1920797877785042297826876342398329981366626138903106707239638623062073330838585770486613528464989802152413220811471053421552758764601275378866508933980136993099
9^9 + 45^45 + 73^73 = 10533405146807286720373659460502060785759379112212598116064998666898426140657790882615617759914831094396745709234759931508925318623262447
10^10 + 33^33 + 64^64 = 39402006196394479212279040100143613805079739270465446667948293404374831811584971639253882284819580176345975173687329
11^11 + 40^40 + 56^56 = 79164324866862966607842406018063266761180441458938437285122402918484170424104310169552877361994067
11^11 + 68^68 + 68^68 = 81589835908549566628948778845927188824021106825908376093331879269943262593090921441573064930996452930496121135091459499137363
12^12 + 16^16 + 79^79 = 817598737071050959409276229318696698168591900537987468276932073768901912096673342793217657607316423968313726492566673678273923566105232874541149775791
12^12 + 22^22 + 97^97 = 5210245939718361468048211048414496022534389576033913164940029913016568215580398296261072019231723279851007241838011659882766685337218633992220688288491655299087016537413082582567277891232916577
12^12 + 43^43 + 84^84 = 435973436827325522360279881406914796368935566412408014666801047266959214000936369697318397346096066940168656241193375655766027231043615387232193494673728498562099
12^12 + 97^97 + 100^100 = 100000005210245939718361468048211048414496022534389576033913164940029913016568215580398296261072019231723279851007241838011659882766685337218633992220688288491655299087016195985205218347720494586192993
14^14 + 56^56 + 83^83 = 1920797877785042297826876342398329981366626138903106707239638702226398027024997104196847572250329782573083762099798980807993002895725961543134834407346088522059
15^15 + 34^34 + 84^84 = 435973436827325522360279881406914796368935566412408014666801047266959214000936369697318397328752293573138388733046233272322611084149603349530457534098877489547247
15^15 + 41^41 + 83^83 = 1920797877785042297826876342398329981366626138903106707239638623062073160162030496354441554188405988281471161451822156686553430667541426768986742630962731453603
15^15 + 44^44 + 84^84 = 435973436827325522360279881406914796368935566412408014666801047266959214000936369697318399379526117133748442366495203539330030558915389368827649274397901769345007
16^16 + 80^80 + 93^93 = 1171963849265444210417582587751425509414623986542768039783389634210694450151344164327247456669487978356117967896050409107202745103300944452206991034477139649315017383181753061046034509
18^18 + 21^21 + 56^56 = 79164324866862966607842406018063254671922245312646690223362402918484176266730674963610409969023301
18^18 + 24^24 + 47^47 = 3877924263464448622666648186154330754898344902677941694531956160410874395482063
18^18 + 33^33 + 64^64 = 39402006196394479212279040100143613805079739270465446667948293404374831811584971639253882284858926584421261711262753
18^18 + 63^63 + 92^92 = 4661010870363696423905966214003100982132353937802439629342577411202087013124250552447236917961362723568709222579851175232314101132993821610968958124002541250261254077596866609283007
19^19 + 53^53 + 83^83 = 1920797877785042297826876342398329981366626138903106707239638623062097517010195519066574031793595215376356982906237443086434589542435116137853459626362057914939
21^21 + 26^26 + 56^56 = 79164324866862966607842406018063254671922245312646690223362409074603756473848639352209401831651653
23^23 + 32^32 + 68^68 = 40794917954274783314474389422963594412010553412954188046665939634971631296546922222423863368437310618080624762599598059186919
24^24 + 39^39 + 56^56 = 79164324866862966607842406018063254784517392774717882763151851908706966131146699513198190533119191
24^24 + 43^43 + 84^84 = 435973436827325522360279881406914796368935566412408014666801047266959214000936369697318397346096066940168656241193375655766027232377351164082477619113893870957619
26^26 + 35^35 + 64^64 = 39402006196394479212279040100143613805079739270465446667948294506753221125645912719320279875550574604423512826057467
27^27 + 33^33 + 97^97 = 5210245939718361468048211048414496022534389576033913164940029913016568215580398296261072019231723279851007241838011659882766685337218633992220817398531743503541344055052909831132881534819018053
28^28 + 59^59 + 64^64 = 39402006196696661278814472355758348506413138794914729578480576128900860152194191752791656089744242832418292148547491
29^29 + 48^48 + 84^84 = 435973436827325522360279881406914796368935566412408014666801047266959214000936370198020475592211612747675413970860482616231636603296987004039670841139738478005261
31^31 + 64^64 + 90^90 = 76177348045866392339289727720615561750424801402395196724001605146963333737512250298641100143613805079739270465446667948293404245738840671341334650224865527944700121384035041247
32^32 + 54^54 + 55^55 = 528286650513762562786451138934901369991153545728533586167866656330073914898729733196007681566407
32^32 + 69^69 + 76^76 = 87464740776733105365396438099869252271646665449763117873625416364159260449821323557849705385587934690388428550525710438009656797528664028836981
33^33 + 42^42 + 70^70 = 1435036016098684342856030763566710717400773837392460666392490150130937545296572485882012252015485653664000503403312775230716396577
33^33 + 63^63 + 97^97 = 5210245939718361468048211048414496022534389576033913164940029913016568215580398524534108366198768259751519579003534060701791407828152463947013890666249058064250446454817827129734022494807468097
33^33 + 80^80 + 94^94 = 297864151605271565671522691888487610082907956052537333166613522312041004445796225209948231096750325129944016829477021503670303547569409565427112857007362085881999202363463800114845336609
37^37 + 48^48 + 50^50 = 8882284899079515782708227892306947026666030124033707486218854203868480905047351845973
37^37 + 58^58 + 84^84 = 435973436827325522360279881406914796368935566412408014666802947573340155595416133581263256723656226994882859353596398184230398370019163876078867133592036419658837
38^38 + 56^56 + 83^83 = 1920797877785042297826876342398329981366626138903106707239638702226398027024997104196847572250329783648995564079792962868422255751849740658622192125755327574347
39^39 + 41^41 + 99^99 = 369729637649726772657187905628805440595668764281741102430259972423552570455277523421410650010128232727940978889548326540119429996770825349677401744263549953448483395812705120907266650190001348405499
39^39 + 62^62 + 86^86 = 23273773687010809805103263055261877739102071580597940410929578754776743489063041366604653798144723500671297357813917754962379668941261341603042947049873470933491354839
41^41 + 76^76 + 82^82 = 8565168191027986598571785581656645742204215247229463095833103436851058447852969472177986666177528487895945859117779903442398758402031772204655766303110894441
41^41 + 83^83 + 85^85 = 100142174126331684536804823936188347206229726834331613365410579745811669866315225666307869547113428836607359228551783278879880523668085257333982264419532757165047353
43^43 + 56^56 + 84^84 = 435973436827325522360279881406914796368935566412408014666801047346123538867799336305160803364159321612090901553840065879128430149527785811336503664217404448437299
44^44 + 93^93 + 98^98 = 1380878341262658714506177247462727309464355354511437784220839352953734795012952144912769658045968061606585338793923209164420614510777066663149388920416939637420392326008424058316632313945617145933
47^47 + 47^47 + 63^63 = 228273036346967044979900512337165530156667551651388179163251165381929227111693938279054637971898106544571908468573
48^48 + 61^61 + 70^70 = 1435036016098684342864068244129256661174837799031396626610207152702165560336919950535676162344468970853217829109132842625712835117
49^49 + 67^67 + 71^71 = 275006373705798627899794653439821885033867103891904078976198493066614790143795102827639062186915317673214405848578954740244507336643
51^51 + 61^61 + 75^75 = 426181657761258833198605424151968794876353861553896334273149939531875976559410524485135689604488287794548629024764307465267700147540336668587
51^51 + 74^74 + 84^84 = 435973436827325522360281985898822381800924184914692357495610164753519335227419109602564502356124773671089873400528310490052829406569433847126444035469921666745963
53^53 + 65^65 + 79^79 = 817598737071050959409276229318703606420756661458838873815650898903149165187530079853924051450132687408571517035288508845239777973686733864536433816917
58^58 + 87^87 + 94^94 = 297864151605271571143886699404293526289042440427466308483421315380144879251775883993328945891334959638296662699335693858802527438435270833155562198388371718825612813504486375316737655463
66^66 + 74^74 + 85^85 = 100140253328453899494506999164337856461680350056679798136769574582135747739332781013021620458928442003554221983442614866066975056662902508987777913433230922504010117
71^71 + 74^74 + 74^74 = 4208984090177237460698612226119734245487631615425800544812861116667077573082138155195608547418829858480630522962165436531602631101602621623

Q1. [a, b, c] = [1, 5, 7]
Q2. With 0 < a <= b <= c <= 200, I found 485 cases:
n [a, b, c]
1 [1, 5, 7] 6-digit prime (666 at position 3)
2 [1, 11, 77] 146-digit prime (666 at position 132)
3 [1, 23, 167] 372-digit prime (666 at position 83)
4 [1, 42, 138] 296-digit prime (666 at position 187)
5 [2, 8, 77] 146-digit prime (666 at position 132)
...
480 [149, 178, 178] 401-digit prime (666 at position 263)
481 [150, 181, 188] 428-digit prime (666 at position 348)
482 [154, 172, 197] 453-digit prime (666 at position 21)
483 [162, 162, 167] 372-digit prime (666 at position 142)
484 [166, 167, 170] 380-digit prime (666 at position 155)
485 [171, 172, 174] 390-digit prime (666 at position 297)

a b c p
1 5 7 826669
3 8 22 341427877364219557396663500827
2 16 27 443426488243037769966696374692859444423
2 16 47 3877924263464448622666648186154330754898344901344205917642344074630570094614483
18 24 47 3877924263464448622666648186154330754898344902677941694531956160410874395482063
37 48 50 8882284899079515782708227892306947026666030124033707486218854203868480905047351845973

7, 5, 1 -> 826669

22, 8, 3 -> 341427877364219557396663500827

27, 16, 2 -> 443426488243037769966696374692859444423

47, 16, 2 -> 3877924263464448622666648186154330754898344901344205917642344074630570094614483

47, 24, 18 -> 3877924263464448622666648186154330754898344902677941694531956160410874395482063

50, 48, 37 -> 8882284899079515782708227892306947026666030124033707486218854203868480905047351845973

There are 63 such numbers for a, b, c < 100.

3^3 + 8^8 + 22^22 = 341427...666...3500827 (30 digits)
2^2 + 16^16 + 27^27 = 443426...666...444423 (39 digits)
2^2 + 16^16 + 47^47 = 387792...666...614483 (79 digits)
3^3 + 14^14 + 68^68 = 407949...666...291419 (125 digits)
2^2 + 8^8 + 77^77 = 181880...666...7774017 (146 digits)
2^2 + 22^22 + 101^101 = 273186...666...666...233689 (203 digits)

***

Adam wrote:

I get the list of a,b,c values that follow, sorted in order by the sum a+b+c. I think for many of these, the 666 is appearing in the really big a^a value, and then the
much smaller b^b and c^c values are just making the sum prime.

[1, 5, 7], [3, 8, 22], [2, 16, 27], [2, 16, 47], [3, 14, 68], [2, 8, 77], [18, 24, 47], [4, 15, 70], [1, 11, 77], [18, 21, 56], [3, 40, 56], [21, 26, 56], [11, 40, 56], [10, 33, 64],
[12, 16, 79], [4, 11, 98], [18, 33, 64], [24, 39, 56]

***