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Is there a natural number $k\ge 1$, such that $11^k+k^{11}$ is prime ?

I checked the numbers upto $k=3000$ and did not find a prime number. On the other hand, for $k=76$ and for $k=142$, there is no small prime factor (http://factordb.com/index.php?query=11%5En%2Bn%5E11&use=n&n=1&sent=Show&VP=on&VC=on&EV=on&FF=on&perpage=200&format=1)

I cannot find a reason that $11^k+k^{11}$ must be composite for every $k\ge 2$ , for example because we can show that algebraic or aurifeuillian factors must exist.

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You did not check quite far enough. $11^{3100}+3100^{11}$

$=2076462013023723087177998371272629862972883202603058638957413939624626$ $9054240698423859652025721171495403183350925185327797086492200207876487$
$2657520023839950330031205883217443601268748782180924528196277373392564$ $4013266704424018036520898747850451175736951267420442305614435735217327$ $3971093935910922095895742597356204629264981808290204113439085537450904$ $1054716017041784246181255107634302837222895447128850530244058819200215$ $2362870017852948178955739528730187590107749130966131061785154952403600$ $7845259931210631021751416234425093201119837728222208324398548386042895$ $9584011040615951740645304512954981656788544230076265875001026816088705$ $6757307230421011598858076933236873935084068306578841134872968149221788$ $0717817364570019141856814901370572392334191752979013227566442886023434$ $7285704153224120229496485704857078154855202352223685401328549708656537$ $2745103719682773351988518415795763107623953506997522673140901829546037$ $8418905329436977478581214760667145942730920343954357670045645928260538$ $8061193029697759164275948364125097680392267910933560309185294496717695$ $7869168427818687080167508439109316989952426208387505064118843391126904$ $8809439180832931991818599346591305891117152763850592098966904948668260$ $7218165257463715929191901113894143802681224509003171286959752701316000$ $5149727764856832739587273643660076429162547313056786072738282051427143$ $9059263411928545834661840831401401793837448294722205096917466151658118$ $6345840545618942101785072630580361383262366193852415870364775543675397$ $7411556835071609937251107708659627426204781277098083989903899975463323$ $0297607332321477653359454339442543504332822307845301255056579651182760$ $3197214280255284847766302464422208909240777526693592272352267547214798$ $9279732186573642952954772173255469021780823659279533971856573468351741$ $3270025882653830663768988149700654876357099836728713717899352117384639$ $9993052359435350597385181227038675446499316142028048344457869772495345$ $1159392734366193006180448767048576470894500056348333701493332319434778$ $5490882030798588653090632863574646380731305190374645481940398446527831$ $0722028247156052753014565407431192576032277984102207926118128755743971$ $8340985455265496911157467144349144537796113479118340686160573603130766$ $1099123424689008589232889055800413648775020397984208864809200556571931$ $1582800146997406161685727978841949295312774927206609369501218544956948$ $8530144248346132022932824472115888270376191468004515998634183646374938$ $7492291650568266714709826604252169619563731647361927100184078247062181$ $5224715926448818703439624866795396110172267449029716976203305026056481$ $8323355342007447572624200323791062810049504835193102542786506344964036$ $2891135921476558034980600392420027840933546664159778908938853495165590$ $8907646835427289592872012218998684082574118686855754880914643032478309$ $8873209783047376068101435630510253377633655787909767034270013933181425$ $6575659583293821949031203842568947299487797641091314053619475074233370$ $0097144671013963445110078855965695128408676810683471873791915379119568$ $5384648967264244558381359201014768419939135316262258913110928604355249$ $1071911512953583221678783961802928432334347315692012480310793727126598$ $7925284867474264761232617448873485010382295276804354264892976999063391$ $0697911628932721420433293413759170876405267637504799091260822608897310$ $259826001$

is prime.

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  • 1
    $\begingroup$ Thank you for the search. I continued and came across this number. Did you proof that the number is prime , or is it "just" a probable-prime ? $\endgroup$ – Peter Jun 22 '16 at 15:04
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    $\begingroup$ math.stackexchange.com/questions/123465/… $\endgroup$ – almagest Jun 22 '16 at 15:06
  • $\begingroup$ Unlike you, large numbers are not my field. It is a while since I looked at the latest tests proving primality with certainty, but I suspect proving with certainty would be quite hard. $\endgroup$ – almagest Jun 22 '16 at 15:11
  • $\begingroup$ That is true. The number has $3,229$ digits , so a rigorous proof will take a while. But the number passes many SPRP-tests, so it is without doubt prime. It is curious, that the number of digits is prime as well. $\endgroup$ – Peter Jun 22 '16 at 15:14
  • $\begingroup$ Ok, you got me hooked. How is that number certified as prime? $\endgroup$ – Oscar Lanzi Jun 22 '16 at 19:08

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