Glad Hobo Express: March 2026

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.

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).

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

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