I may have stumbled upon a very small number (N) (really, almost negligible) that behaves in a rather suspicious way when you Collatz it.
Here is what it does.
It starts by going up 2 steps, then down 1.
Then up 3 steps, down 1.
Then up 5 steps, down 1.
Then up 7 steps, down 1.
Then up 11 steps, down 1.
Then up 13 steps, down 1.
Then up 17 steps, down 1.
… and so on.
In other words, its Collatz trajectory politely enumerates the primes:
2, 3, 5, 7, 11, 13, 17, 19, 23, …
I don’t know how long this goes on. My machine gives up long before the number does. I have been told this is “normal”, “expected”, and “2-adic nonsense”, but I remain unconvinced.
Some people suggested that if I let it run long enough, it might:
- discover new Mersenne primes,
- encode the Riemann Hypothesis,
- or at least heat my phone enough to make coffee.
Others reassured me that it will, of course, eventually fall back to 1, like all numbers do.
But who knows? Maybe it keeps climbing forever, calmly spitting out primes, one by one, unimpressed by our hardware limitations.
Unfortunately, my computer is not powerful enough to settle the matter.
If anyone has a spare supercomputer, a few spare centuries, or divine insight into the Collatz map, I’d be happy to collaborate.
Until then, I’ll just assume this tiny number is very busy doing number theory.
Here is that number:
N= 28060447227691927590448267598862779614394535296743894612926453547144284219523789279612059939347649868142482789596559449180759745182694115786927979573023090008004039921443223336556383724060477643648409904948481625496859269816239340371675520360868775854524633140268842479213290757087194372879080275214265216710267369276463528751470120914113257203400968222401171376135398368236340440718331896388997498614095570230469479443071568609168677955777760483345399014352521306891787407003012690102319763302179365294869620859759812409683851078484796314321847319950337830273618875895409536521960793587553065944120994687650473417931327381355971575449505048433034467786765041382843790311481393771303003472051186351507478298947978436022194677403669812577246595319559200469531108519277073222957156406813571799898791242290827433771106699047929644780560760384826112305990471852739295188463126719584773818083148795890242797490342755071343852063133617908906721741911438832787694822038406173856519378847251656860311006880261280055756398364833615502238244247047515925243487028189071525442775401461106360226929020207632889589652197778232857589518818213229734517461058638928132265638487318666303979256680266333232579818428195718874997977669456823262441174283868434368499496237503896030842281493599331008271379664287916507320271321101423842784766527839748262443961548596256697778701239740814540495625632037386011940817537795914148645996470862023014267681252909785865327448148518336580771829544476939386034715683388591454210923511613568755735711375332923456380570586090651294007964624782081863697135952044545674710560532847902685955310001110802471894258344726100773585587037908258967439592382694482815002882914475416955279279433182018334120096074819393709575852603683196119399805057276743368300722752204833000168791852001422604525253304197822591519912236226037243931106150941543518934664743975374985720124712530831378634302061259568239899915224712981624239568346719527879138103582999760723024992149449379566065099673784151605623687357679190642208924331712339606819346056567409554340348160229488437727282016572607453615463525309306878356136004567261020040973839592968767445320186077507914187199913603189446253032836268434133497552200755866459665719473142500761223529119311400141896815677642977696694287953552170114685059818024270529155854295071917000324251683236861472375510179965431413488400649235953545699079950516341841510551991862062650075245554708349847008327079844879417008900232590374760528731857795445116531517067979401119560418153220700659998723866762547272987178611675430578674315514103607045042487597784938389595945543355281686573497610928691074823300867807806719603399906091381861355096096827639066555114616241278049595977939305236551212948505998755448597576029995422307597663818271711341822615867686382462730051418072700449065007167293455248979522312922645915026321202911580664223016485379951239158627764510627797380678122356039154075759064968228581060964939925704791431124578157663535627977317301879864799293733173265519345333015653644863538540434897886706604848732219510943284449605381298072985778100557898361608082083477310988396987123295241349508801191169570449110368772981821477116719073956615367236434645950707108967002463656902737871061315166469715510256940008302498234753529697053498151320584510541008992121496002043576139170197485703826943104481349871098038105602818974661397686320541296871923396746449929743330810020784053008180772018018923238989369148890914308284863499015040590158660552141973367714576229616835907284176031046935103015399640018867679834845686268479378203065117831535950005968464406285906595452163106131056705482527121787023557265832937023438390926010312921328103344759834923753355783853937509439760120070322983859134928257785086641844116883581775220779963209823460849205103455556515917386234049872322972966342771960988040513923435998781050758167625193098654543159621906798397329381531618137843333876787789227656065219036403696370634914116021666941146446877805809566558701730215106142015651415125977686998449716903298917827503752053601736838435830066641944218568843204249496814090490441712382014947434085065475005800098217158518685485099420640102552427705605628667259251285240540999888915018005033415831630571015623713783196753965173125509555743656655422707682846184906117432673796986161282212922579450694061986193696086915993376668760970549675296237772294325867277795189500172072139238275550975059463151981818093215260951412045704491819236787173322343653994462042453210335095738454879780214895627275575742174406529148489870465276182082938749757859695806726373884964520194422091378440216646206947697956144258296063511238755970858565284291156465021733325138409820244535557180867564729356323931869897789779306520907747873603263364158785502372022920524789543672555836324104910774544355509017595509212151608204473769971129704010430878615956672344966249226954915212008195789890728554976388067889375936124825949062256976639399125905241627185075012871047146644181217561234351965351198117692580610099725172624422910571600767563777151107374259179836082660334640218755727211106396544308964301768192416273802432394998302191753613384990577731840739961128368539958350730447988415951301774134358489863300866942950532600632538159048823383109148248346671765887819605576679220094460792601128401039293068837936292419374801421630202988913131133789450863777511325189968871551044101166938078767111862983716329387557106017419671179230542462658484750992861556033880641004946662302251531280907910130817300116121176429941414332100280334225546641345372205544454088963506372756408884891647307097291366349451133828996416963121391227970567720934910969174353452578623796556457265524346211537498759519565641325089138272778287877865376455643484784708657790553750504659218581245584103596194682814236826001630799723514571818044389470519275881291964134366048148902881966736147900995767283140702010804176859741224501792992755269377754635870236404535153526330485789513089433846649841946022512000092996403380086467441384589291505238143967459851950432449520167555724893889734599696821938652972086087913245657905848387308382785924517058760413908960215187078882423079063636588099160125507606606322834159534738267457651558514300137433234066627924102323947823550867168058745790205111245185334948923946316820615116663530630936712498377711312061232221812852787870221540177518830274398933023465450008340852491873712452287382814991861365875433889693466522588098327996279809448145462119643352626996175330347775056147136158525825240712421247887222875