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

status: PRP / PRP

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, bringing about (of course!) a flurry of new records:

Gamer "Gelly Gelbertson" found 10^10056+10^6692+10^5872+1 [p] (a 10057-digit term in OEIS A393530) two days later & Vishwath Ganesan, 10^20000+518406362*10^9996+1 [p] (a 20001-digit emirp) two days after that.

Update (March 19): Serge Batalov has found a near-repunit 10^22822-10^63-1 [p] emirp!

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

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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

The five-millionth term of A157711

The sum of the first 4006 prime counts of A383675 is 4998755. The sum of the first 4007 prime counts is 5001521. Hence, the five-millionth term of A157711 is a 4007-digit integer:

4998756 10^4006+10^39+10^18+1

4998757 10^4006+10^104+10^53+1

4998758 10^4006+10^108+10^16+1

4998759 10^4006+10^177+10^120+1

4998760 10^4006+10^192+10^59+1

...

4999995 10^4006+10^2671+10^2184+1

4999996 10^4006+10^2672+10^696+1

4999997 10^4006+10^2672+10^1708+1

4999998 10^4006+10^2673+10^392+1

4999999 10^4006+10^2673+10^530+1

5000000 10^4006+10^2673+10^876+1

5000001 10^4006+10^2673+10^1124+1

5000002 10^4006+10^2675+10^1928+1

5000003 10^4006+10^2676+10^1232+1

5000004 10^4006+10^2677+10^1160+1

5000005 10^4006+10^2677+10^2092+1

...

5001517 10^4006+10^4003+10^3570+1

5001518 10^4006+10^4003+10^3818+1

5001519 10^4006+10^4003+10^3854+1

5001520 10^4006+10^4004+10^609+1

5001521 10^4006+10^4005+10^1892+1

A157711(1*10^6) = 10^1793+10^673+10^615+1 [2025 June 19]

A157711(2*10^6) = 10^2535+10^1160+10^398+1 [2025 July 21]

A157711(3*10^6) = 10^3103+10^2747+10^859+1 [2025 September 10]

A157711(4*10^6) = 10^3583+10^3040+10^2776+1 [2025 December 7]

A157711(5*10^6) = 10^4006+10^2673+10^876+1 (above)

A157711(6*10^6) ~ 10^4388

I'll continue to update (here, until the next millionth is reached) my current plot of A383675:

A383675 to n=4166; max=(4137,3448) [updated March 16]  click to enlarge

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