A Long Way Home

With large composite numbers, the path to a home prime can be a long one. This is certainly true of today's composite number 20240802 corresponding to today's date of 2nd August 2024.

$$20240802=2 \times 3^2 \times 863 \times 1303$$ I wondered why the algorithm running in my Jupiter Notebook was taking so long to calculate the home prime for today's number. It turns out that 51 steps are required before a home prime is reached. Here's the progression (toward the bottom of the list the larger numbers may run onto the next line):

20240802, 

2338631303, 
2953791951,
311233891689,
33741160974887,
311088424547577,
3347298836142697,
46728012558965891,
64269417270645951,
3713131313117863595291,
73131473868929224040761,
7157142964517218506685541,
37461590937100697383704241,
676775575541914812748937421,
33370837121353850660547143863,
103512102850518963266335968989,
127815055927956842230443590307,
3333334978717647771371995485881457,
35545734763423801200354273504161619,
3115096476834788845097622093725646027,
3171315807287116497834303893733982641037,
1938593133169178678118176191414687586788699,
7806127981924834247371605710953994219337521,
1113109500810161155118640365150812276512107483,
79263561175470733867058721787851202039247470829,
199350779397608484767800851518411783020176720951,
313893271587387422969396469847860991092333976542173,
19916894174850252716742066231437827881452587132198347,
333185411566209479042084941629363924712738008623074892523,
3223494969430572079429042941294041931241139987441774025309,
32950793229023376852456919797337294607377341857102318172479,
301958427353613415282754723578180598828447843204448262081363, 43671933129241050741953825012470051854111324175024183278643133, 3713710348596620298325276499784604740144926044521019175560918133, 43432033318665629558961303257652524300665434496179740124768638737, 438318191912142630256950384088336721700875079874885901277773615661, 33952660372710397050719260552116633226209979280798089039448973313459, 235473639043239560844701954001988841885040187355225129838688185185493, 333671045063366301288318623135523361878429909102403826971796025548984839, 7415772280289158642876806700567063245584286858045263726536419020786144949, 31915460753992647178331840969114638386529105925505411017314020817595444429481, 388215755079417212095611247626509474236722043804223179439175841269325900221647, 43119489467805082469367345227763064513316151461945387396217629252808742563363617, 7178040010219740423774713801327263673054482930504828408891656467662033385134241117, 109316473995628413986454085350966468455812414207316882972860619920620319817943453409, 2323834579133836345413046351762997943556241757630498282284882868364567485130261890219, 317300267606964982136843437711517492491619282822669294694863577107023014585554599905739, 52328463255749224374714279838161134823233769307043696613665389077650729417404884413292271, 3714392062034766290224112375598877216263278646829696160089485291757181880256559477546163399, 3394732071391164431104598827162026435326822795676460127579388576883851144072368157718797931641083, 19294447233994807147867590543139812872007673327880662223899101540689566880289506377715917567105333859
7772349034187438087121221820512743165100759812317447221713249182984386976343967857556366241938724210083

That final number is indeed prime while all the previous numbers back to our starting point of 20240802 are composite. Contrast this with yesterday's date number, 20240801, where:

$$20240801=7 \times 2891543$$ The progression involves only two steps and is: $$20240801 \rightarrow 72891543 \rightarrow 3191278799$$ The day before yesterday only six steps were needed for 20240731 to reach a home prime. 

I've written about home primes in earlier blog posts including Home Primes on 2nd May 2021. There are 471 numbers less than 10000 for which there are no known home primes with 49 being the first. It would seem likely that there are many composite numbers without home primes amongst the numbers generated from the daily dates.