A random 5000-digit emirp pair

Print[DateString[]];
c=0;While[c++;
PrimeQ[IntegerReverse[r=RandomPrime[{10^4999,2*10^4999}]]]==False];
Print[{DateString[], c}]; r

Tue 24 Mar 2026 13:20:15

{Thu 2 Apr 2026 14:16:15, 1459}

13734728109222387308771911883098209504266491220378415449228996256958556204646209255426997204731352359449756449041282911838827753960110717667741546300796170994606405370205525990467395376009947818863867098443204707446280728853460347596959087028624624873124154773387129505731773658548559945072217868331005644146148486987796403656801876947383547342120251584407014371145256380254312563183866535417598667871824487793918102277168404192702988592723616031279706185729027937193115459363751801100834619648559534755862280622532330328020545564913373716599937718339772157065308727535050239049475309383701181923039664677398022448040478292058445782572749973954471541202254146634154863136377224819621843776574351493423099417717302803582947518052391674365890421676801423682263895436033477349103616708548394962338144244739840618759795636515556851999913235196980652229399132598337576053168937606823091007475684552339301961892031641961714248878071017023578324887088755960213127771073546703472455657291745675975849520494469065429206686852364253408177225968538981379921456249713012719550869982628800215250141536070000956939408031634199798833269213339776024607090221636520868945020007583404435400304984816689218801098149473216367358358482560919964155113665854879162787733470212782577499299460614936316249501982462197582182962661480246316492791906606967203513849810898912521723364783300800397527161480006666132471635152505857552637781618190671943929866125374286353526295866879589156528069268446360659588156073182367126878655885846017841827594449729142868018298455042670803762560261034102071781850653996720652827866216113764531977499870831481877699820219265369589791816326483849854098097404115191627985774704862207480350650524253157202313714981369031023817964668976804563887152280560591825099628968174198256928594455474900614206201208779564019513967006148424961575107554035816503265109246522149031864758498683614882701625685700872773227021642782062868941145310351815390646299885415571562126969030978677691992248131837773010974896587969945625215247802998599555149844869576921906654915655378310960741531349433956427372335169637595119451916257126526280015229284620103272436514641785833667906959477216083771551052106172820461661921196270515411038383894689853536962945436545227700069617650926512319248691302624136735627062344420181527288419813974095715282212691583319754787733622431421874596766566781210035118902671019438169295573224505091585397715351315220211971040233736523974341495533949868364720277377782599728157350029382463638680674321687547356330765388203811846145371779763392852911636547647481956184757017178391537543535061700390946825273445872542682841891167992835988152856822682714317459393194812472058333096943727715717938929497658790667620293930654794830364719577049887208648521075972113202459509425634726291406990514668964980723429065928682562597336923568312002853777432281023039930293635129199440948687280466516218157232869951411288874973458368152724148075445625803015700585387849533731143257417116714188837383227808909962194283481216500073402466043334005782482811768333556062299980861037631738960230903988141466474914973109939569029008994447469841264687420328226975199672037910132356689256318605922099489971249725724478498782743882782690449251101947062584707755445477591011686309629814598618824681005820262188651690555030066130820115706496744154662972097838464620864109248150554977389731984069894739940147765980248175151267385448245010741453588017824020636560200828952438588646996716310593319136256633679956860537422448902141226501877981016117706838811826647919831942614642062404688464031914722034158404047771516715512437734236041562161536854739816352115170865325632600798156951738783115882250368309347812703075264217063439654304694709966393200029975921707823139029998209103051390416799743854828992885437521880547367118888862322413374719343146756530741790302078652676659228463250342071145099375265620764389943047611045428183649445593679065081627297218931027072918317542023094964352985593971204351149921620655199890626051131805882424498700115214465774587948209887902292657524952455402231405821109075638114959572631886949305122157028523531876233977699646931396075719474191681597682924071799804008775563101404661918081796360660189113306106394092364326877126214067888878532744860157995659689449232748995366653945094234108364647434237692992245404080070494425022553082178108569582011506349785726297495243949030211388665827342268742694968604680029493370744659894200765912072030956961758839895603163676944756552653917173731760257052296666264677111508448519434784124629839512816675234329603341834304686287860364334731171629288773512175476941081385777926407651284584960519842623921169707152430953904143362518957259889371822834194124362865942682637039390381181943090487239006104649608076772200752347770219391585047496141065719922022050533141603543274273449631439746074641670898190472653657991251980586966500792504004728827518312237017772593842641588253316664038908417318413936902013666661226800919070758421595913214002598621253

IntegerReverse[r]

35212689520041231959512485707091900862216666631020963931481371480983046661335288514624839527771073221381572882740040529700566968508915219975635627409189807614647064793413694437247234530614133505022022991756014169474058519391207774325700227767080694640160093278409034918118309393073628624956826342149143822817398895275981526334140935903425170796112932624891506948548215670462977758318014967457121537788292617113743346306878268640343814330692343257661821593892642148743491584480511177646266669225075206713737171935625565744967636130659893885716965903027021956700249895644707339492008640686949624786224372856688311203094934259479262758794360511028596580187128035522052449407008040454229929673243474646380143249054935666359984723294498695659975106844723587888876041262177862346329049360160331198106606369718081916640410136557780040899717042928679518619147491757069313964699677933267813532582075122150394968813627595941183657090112850413220455425942575629220978890284978547756441251100789442428850813115062609899155602612994115340217939558925346949032024571381927072013981279272618056097639554494638182454011674034998346702656257399054117024305236482295667625687020309714703565764134391747331422326888881176374508812573458829982845834799761409315030190289992093132870712957992000239366990749640345693436071246257030721874390386305228851138783715965189700623652356807151125361893745863516126514063243773421551761517774040485143022741913046488640426024641624913891974662811883860771161018977810562214120984422473506865997633665263191339501361769964688583425982800206563602042871088535414701054284458376215157184208956774104993749896048913798377945505184290146802646483879027926645144769460751102803166003055509615688126202850018642881689541892690368611019577454455770748526074910115294409628728834728789487442752794217998499022950681365298665323101973027699157962282302478646214896474449980092096593990137941947466414188930903206983713673016808999226065533386711828428750043334066420437000561218438249126990980872238373888141761171475234113733594878358500751030852654457084142725186385437947888211415996823275181261566408278684904499192153639203993032018223477735820021386532963379526528682956092432708946986641509960419262743652490595420231127957012584680278894077591746303849745603939202676609785679492983971751772734969033385027421849139395471341728622865825188953829976119814828624527854437252864909300716053534573519387171075748165918474674563611925829336797717354164811830288356703365374578612347608683636428392005375182799528777377202746386894933559414347932563733204017911202251315351779358519050542237559296183491017620981153001218766566769547812413422633778745791338519621228251759047931891488272518102444326072653763142620319684291321562905671696000772254563454926963535898649838383011451507269112916616402827160125015517738061277495960976633858714641563427230102648292251008262562175261915491159573696153327372465933494313514706901387355651945660912967596844894155599589920874251252654996978569847901037773813184229919677687903096962126517551458899264609351815301354114986826028724612072237727800758652610728841638689485746813094122564290156230561853045570157516942484160076931591046597780210260241600947455449582965289147186982699052819506508225178836540867986646971832013096318941731320275135242505605308470226840747758972619151140479089045894838462361819798596356291202899677818413807899477913546731161266872825602769935605818717020143016206526730807624055489281086824192794449572814871064858855687862176328137065188595606364486296082565198597866859262535368247352166892934917609181618773625575850525153617423166660008416172579300800338746332712521989801894831530276960660919729461364208416626928128579126428910594261363941606499299477528721207433778726197845856631155146991906528485385376361237494189010881298661848940300453440438570002054986802563612209070642067793331296233889799143613080493965900007063514105251200882628996805591721031794265412997318983586952277180435246325868660292456096449402594857957654719275655427430764537017772131206955788078842387532071017087884241716914613029816910393325548657470019032860673986135067573389523199392225608969153231999915865551563659795781604893744244183326949384580761630194377433063459836228632410867612409856347619325081574928530820371771499032439415347567734812691842277363136845143664145220214517445937994727528754485029287404084422089377646693032918110738390357494093205053572780356075127793381773999561737331946554502082303323522608226855743595584691643800110815736395451139173972092758160797213061632729588920729140486177220181939778442817876689571453566838136521345208365254117341070448515202124374538374967810865630469778968484164144650013386871227054995584585637713750592178337745142137842642682078095969574306435882708264470740234489076836881874990067359376409952550207350460649907169700364514776671701106935772883811928214094465794495325313740279962455290264640265585965269982294451487302219466240590289038811917780378322290182743731

Click on either number to check its primality.

A random 2000-digit emirp pair

Print[DateString[]];
c=0; While[c++; PrimeQ[IntegerReverse[r=RandomPrime[{10^1999,2*10^1999}]]]==False];
Print[{DateString[], c}]; r

Mon 23 Mar 2026 16:19:04

{Mon 23 Mar 2026 18:37:09, 305}

14855218387530938731264800899806518554251409448753318067281617721104517975019660570816502191121802141387619791604557638640603077054520997318307203718761757739257990957393586487037638214729490346907077085366489735302991920794042479840185960548800113792237378778189700214336498150535494706550605947647886952709343445154850786791334376804237058351294799668452160206131345032141325576891497636422462012076224432607954324056613347607288129441043897934814166919553140730183934337952784895386915184278773673374499327984879217657567333750360009291505056111578053084511773793397248168145190194638081779158080985373127557284411266286462358754778798450217253924795704562031795767586150594858361787012654211320033792723983297048174107728830987370130777266997038517867493302588134280307951436792574170079530221417396081625730302595728237532318481405855540975409890906976902320267307459218191770761431037983144020890302946754575512111129239578508080696921714584513835630889334839362187530001616420174589110874531666851407830633657753457085270606608334710585505853591828745779378101187982074326317494916222176946818501389870085531903573860851057307085813978075441446820544507426572075052156092931837527279166150509917972776167153657340031646179631733906184205918255895475754364175238057482645899534235102077183676796749323690141704894319808779172119291942162868677765137547565809194418652476017816016763486268007290212946536413065587272480288193061788482406655351702586154143963745149534333768316994754590468906962685037527510130840705726077739498669858651281009519862797180044407205030051742292526137434721833180012034655794438599828098375237119761007728650703770787960212849241788574933755087711157262236337487014829313684940715849983628697605605634086992019999606613109048737110152003061306410080888361354585657807374315692388968946536330992803388106574848702486355828678460394253967147711575515270818557921500146950778382811866201918398021331397625888200120115790700195112386497426010970976957364112704151652043

IntegerReverse[r]

34025615140721146375967907901062479468321159100709751102100288852679313312089381910266811828387705964100512975581807251557511774176935249306487682855368420784847560188330829903363564986988329651347370875658545316388808001460316030025101173784090131660699991029968043650650679682638994851704948631392841078473363226275111778055733947588714294821206978707730705682770016791173257389082899583449755643021008133812743473162529224715003050270444008179726891590018215685896689493777062750704803101572573058626960986409545749961386733343594154736934145168520715355660428488716039188208427278556031463564921209270086268436761061871067425681449190856574573156777686826124919291127197780891349840714109632394769767638177020153243599854628475083257146345757459855281950248160933713697164613004375635176167727971990505166197272573813929065125057027562470544502864414457087931858070375015806837530913558007898310581864967122261949471362347028978110187397754782819535850558501743380660607258075435775633603870415866613547801198547102461610003578126393843398803653831548541712969608080587593292111121557545764920309802044138973013416707719181295470376202320967960909890457904555850418481323573282759520303752618069371412203597007147529763415970308243188520339476871583079966277703107378903882770147184079238932729733002311245621078716385849505168576759713026540759742935271205489787745785326468266211448275572137358908085197718083649109154186184279339737711548035087511165050519290006305733376575671297848972399447337637787248151968359848725973343938103704135591966141843979834014492188270674331665042345970623442267021026422463679419867552314123054313160206125486699749215385073240867343319768705845154434390725968874674950605560749453505189463341200798187787373229731100884506958104897424049702919920353798466358077070964309492741283673078468539375909975293775716781730270381379902545077030604683675540619791678314120812119120561807506691057971540112771618276081335784490415245581560899800846213783903578381255841

Click on either number to check its primality.

A random large emirp pair

One might think from my previous post that large base-ten emirps congregate near — or at least involve —  powers of ten. Well, record ones certainly do but that is surely an artifact of the convenience of searching for such integers in those locations, expressing them without having to show all of their digits, and even proving their primality.

I recently found that Mathematica has a RandomPrime function which can be configured to generate primes with a specific number of digits. By repeated application of it and checking each against the primality (most often lack-of-primality) of the integer created by reversing its decimal digits, I can now create random emirps.

While[PrimeQ[IntegerReverse[r = RandomPrime[{10^999, 10^1000}]]] == False]; r

3279288520829477235642879267287999159015788532533738855066046440495522398413879852102369701530100869580838076083149157548206285059151203914266414912431580164820025933969565023665242290427065104497454406152915882315883324964117545072539206755054402862798019415649139491569247916954175149446689650326543522910485350967102290630555006513883295579511863580034048864302120190250042483217978026251237207016901362308898161048199473494827584506196053490289923397647958362118557321048941198888848619470770742015547396263817662528411868482723797066437421483218035169802940903987225108903574471943089572864902145297776096342849868617007993544349744145780223402039236076990018520939944400626427281300176781542771736168164872520396475344293444882067084862153824121702601618059184428161120858343524815906494130051717507462817874659446095816309231120150861019479629358822063445459770400310760092233488204681637298530785169163900358943079863912831713129072815997830914582590098549219524593979172161769855800664064007

The reverse of this is:

7004604660085589671612719793954259129458900952854190387995182709213171382193689703498530093619615870358927361864028843322900670130040779545443602288539269749101680510211329036185906449564787182647057171500314946095184253438580211618244819508161062071214283512684807602884443924435746930252784618616371772451876710031827246260044499390258100996706329302043220875414479434453997007168689482436906777925412094682759803491744753098015227893090492089615308123841247346607973272848681148252667183626937455102470770749168488888911498401237558112638597467933299820943506916054857284943749918401618988032631096107027321526208797123842400520910212034688404300853681159755923883156005550360922017690535840192253456230569866449415714596197429651949319465149108972682044505576029352705457114694233885132885192516044547944015607240922425663205659693395200284610851342194146624193021519505826028457519413806708380859680010351079632012589783148932255940446406605588373352358875109519997827629782465327749280258829723

I did not think that I would be able to prove their primality, but factordb (click on either number to see its evaluation there) apparently has some "elves" who download the smallest probable primes in the database and run deterministic tests on them. I guess I got lucky.

Some recent large emirps

Let it be understood that all emirps come in pairs, say (p, q) where the number of (decimal) integer digits of p and of q are identical, but p < q. Since, in the following, we are dealing with record large integers, I will explicitly state the value of q, the larger of the pair, followed by a linked p in square brackets.

In 2007, Jens Kruse Andersen noted the 10007-digit 10^10006+941992101*10^4999+1 [p] as the then-largest-known emirp. Eighteen years later, Stephan Schöler managed to up this by four decimal digits with his 3867632931*10^10001+1 [p]. One month ago today, this was highlighted in an episode of Numberphile, precipitating (of course) a couple of new records:

Two days after the video, gamer Gelly Gelbertson found 10^10056+10^6692+10^5872+1 [p] (a 10057-digit term in OEIS A393530). Another two days and Vishwath Ganesan discovered 10^20000+518406362*10^9996+1 [p] (a 20001-digit emirp). As Vish's record was not initially noted beyond his PrimeGrid Discord chat, it created, unfortunately, a large number of claims of record that were (being fewer than 20001 digits and coming after February 20) not actually records. That included a 10069-digit (still not proven) random emirp made possible by AI.

Updates

March 19: Serge Batalov captured a near-repunit 10^22822-10^63-1 [p] emirp.

March 26: Justin Auger posted 10^39988+5417497422*10^19990+1 [p].

March 27: Serge Batalov posted 10^47253-22*10^21607-1 [p].

Email me if you spot any errors or have something to add.

Within You Without You (cover)

War (on Iran). What is it good for?

Numerical relativity

The Azolla event

Kids dancing

Emirps in A157711

After watching Matt Parker's February 16th Numberphile video on Stephan Schöler's largest currently known emirp, I wondered of course if any of the primes that are the sum of four distinct powers of ten (A157711, on which I have been working now for nine months) were emirps. It turns out there were a healthy number and I decided to create two new OEIS sequences for them: A393530 and A393531.

While more than half (533) of the first 1000 digit-lengths had no solutions, the others had anywhere from 1 to 7 (digit-length 41) solutions. Moreover, the accumulation of emirp-pair counts showed a steady rise, giving me hope that this would continue for larger digit-lengths:

emirp-pair count accumulations for the first 1036 digit-lengths (click to enlarge)

I will see if I can increase my counts to 2000+ digit-lengths. I was going to attempt a search for a record large pair but gamer "Gelly Gelbertson" beat me to it with his Numberphile video comment (10^10056+10^4184+10^3364+1, 10^10056+10^6692+10^5872+1), apparently made on February 18th (although I discovered it only on March 14th):

click to enlarge