Problems & Puzzles: Puzzles

Puzzle 1194 826669

826669 Is the smallest prime of the form a^a + b^b + c^c containing the beast number. [Post]. See the following Curio.

Q1. What are the a, b & c values, for 826669?
Q2. Find 5 more like these.


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

Jeff Heleen and Adam Stinchcombe

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Michael wrote:

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

 

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Rebert wrote:

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)

***

Emmanuel wrote:

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

***

Gennady wrote:

a, b, c  -> sum
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.

***

Jeff Heleen wrote:
 

For puzzle 1194 I have 6 more solutions. I used a short Python program
to generate a list, then used YAFU / aprcl batchfile to test for primality.
 
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]

 

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