Is there a fraction in which the decimal representation never terminates nor repeats? Is it a repetend greater than 9999?

Question

I am new to math and am trying to figure out something I discovered in researching Moody's pilot wave. I won't explain you how I was led to this just yet I want to know clarify a question on math. I am still learning the basic rules of math and definitions of terms.

If you divide 1/65537, is that a rational or irrational number? Should the decimal result terminate, or does it repeat infinitely, and if so, what is the length of the repeating portion? Can this be known today?

My conclusion and research leads me to believe it does not repeat. If it does repeat, the repeating portion is beyond 9999 digits. 65537 is the last fermat prime.

I tried to comment over here:

The longest repeating decimal that can be created from a simple fraction

But I couldn't cause I just signed up and it was sort of off topic anyways.

I read this and couldn't understand how it applied to my situation but I am sure it does somehow... lol.

Is this fraction non-terminating?

All right - here we go, here's my proof:

https://www.calculator.net/big-number-calculator.html?cx=1&cy=65537&cp=9999&co=divide

1/65537 =

0.000015258556235409005599890138395105055159680791003555243602850298304774402246059477852205624303828371759464119505012435723331858339563910462792010619955139844667897523536322993118391137830538474449547583807620122983963257396585135114515464546744587027175488655263438973404336481682103239391488777331888856676381280803210400231930054778216885118330103605596838427148023254039702763324534232570914140104063353525489418191250743854616476189022994644246761371439034438561423318125638952042357752109495399545295024184811633123273875825869356241512428094053740635061110517722813067427560004272395745914521567969238750629415444710621480995468208798083525336832628896653798617574805071944092649953461403482002532920335077894929581762973587439156507011306590170438073149518592550772845873323466133634435509712071043837832064330073088484367609136823473762912553214214870988907029616857652928879869386758624898912064940415337900727833132429009567114759601446511131116773730869585119855959229137738987137037093550208279292613332926438500389093184002929642797198529075178906571860170590658711872682606771747257274516685231243419747623479866335047377817110944962387658879716801196270808856066039031386850176236324518974014678731098463463387094313136091063063612920945420144345941986969192974960709217693821810580282893632604482963821963165845247722660481865205914216396844530570517417641942719379892274592978012420464775622930558310572653615514899980163876893968292720142820086363428292414971695378183316294612203793277080122678792132688405023116712696644643483833559668584158566916398370386194058318201931733219402780108946091520820299983215588141050093840120847765384439324351129896089232036864671864748157529334574362573813265788791064589468544486320704334955826479698490928788318049346170865312724110044707569769748386407678105497657811617864717640416863756351374032988998580954270106962479210217129255229870149686436669362344934922257655980591116468559744876939743961426369836886033843477730137174420556326960343012344171994445885530311121961639989624181759921876192074705891328562491417062117582434350061797152753406472679555060500175473396707203564398736591543708134336329096540885301432778430504905625829683995300364679494026275233837374307643010818316370904984970322108122129484108213680821520667714420861498085051192456169797213787631414315577460060729053816927842287562750812518119535529548194149869539344187253002120939316721851778384729236919602667195629949494178860796191464363641912202267421456581778232143674565512611196728565543128309199383554328089476173764438408837755771548896043456368158444847948487114149259197094770892778125333780917649571997497596777392923081618017303202770953812350275416940049132551078016998031646245632238277614172147031447884401177960541373575232311518684102110258327357065474464806140043029128583853395791690190274196255550299830630025786960037841219463814333887727543219860536796008361688817004135068739795840517570227505073469948273494361963471016372430840593863008682118497947724186337488746814776385858370081022933610021819735416634878007842897905000228878343531135083998352075926575827395211865053328654042754474571616033690892167783084364557425576391961792575186535849977875093458656941880159299327097670018462853044844896775867067458077116743213757114301844759448860948777026717731968201168805407632329828951584601065047225231548590872331659978332850145719212048156003478950821673253276774951554083952576407220348810595541449868013488563712101560950302882341272868761157819247142835344919663701420571585516578421349771884584280635366281642430993179425362772174496849108137388040343622686421410806109525916657765842196011413400064085936188717823519538581259441231670659322214932023131971252880052489433449806979263622076079161389749301921052230037993805026168423943726444603811587347605169598852556571097242778888261592688099851992004516532645681065657567480964951096327265514137052352106443688298213223064833605444252864793933198040801379373483680974106230068510917496986435143506721394021697666966751605963043776797839388437066084807055556403253124189389199993896577505836397760043944641957977936127683598577902558859880678090239101576208859117750278468651296214352197995025710667256664174435814883195752017944062132840990585470802752643544867784610220180966476951950806414697041365945954193814181302165189129804537894624410638265407327158704243404489067244457329447487678715839907227978088713245952667958557761264629140790698384118894670186306971634343958374658589804232723499702458153409524390802142301295451424386224575430672749744419183056899156201840181881990326075346750690449669652257503395028762378503745975555792910874773028975998291041701634191372812304499748233822115751407601812716480766589865266948441338480552970077971222362940018615438607198986831865968842028167294810565024337397195477363931824770740192562979690861650670613546546225796115171582464867174267970764606252956345270610494834978714314051604437188153256938828448052245296550040435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1

Answer 1

If you divide by a number, which is not a multiple of 2 or 5 you always get a periodic number. If you divide 1 by 65537 there are only 65536 possible remainders , so once you hit the same remainder the second time, the digits repeat. To convince you try it out with 1/7 or 1/17 the maximum period ist 6 or 16

Answer 2

Reciprocals of primes; the "exponent" is the length of the repeated factor. As soon as $p > 10$ there is at least one $0$ beginning the repeated part. Note how long the repeated part can be, even for these small primes

I checked, for 65537 the repeated part is the maximum possible, 65536 lenth, including four $0$ at the beginning.

 prime:  3  exponent:  1  repeated:  3
 prime:  7  exponent:  6  repeated:  142857
 prime:  11  exponent:  2  repeated:  09
 prime:  13  exponent:  6  repeated:  076923
 prime:  17  exponent:  16  repeated:  0588235294117647
 prime:  19  exponent:  18  repeated:  052631578947368421
 prime:  23  exponent:  22  repeated:  0434782608695652173913
 prime:  29  exponent:  28  repeated:  0344827586206896551724137931
 prime:  31  exponent:  15  repeated:  032258064516129
 prime:  37  exponent:  3  repeated:  027
 prime:  41  exponent:  5  repeated:  02439
 prime:  43  exponent:  21  repeated:  023255813953488372093
 prime:  47  exponent:  46  repeated:  0212765957446808510638297872340425531914893617
 prime:  53  exponent:  13  repeated:  0188679245283
 prime:  59  exponent:  58  repeated:  0169491525423728813559322033898305084745762711864406779661
 prime:  61  exponent:  60  repeated:  016393442622950819672131147540983606557377049180327868852459
 prime:  67  exponent:  33  repeated:  014925373134328358208955223880597
 prime:  71  exponent:  35  repeated:  01408450704225352112676056338028169
 prime:  73  exponent:  8  repeated:  01369863
 prime:  79  exponent:  13  repeated:  0126582278481
 prime:  83  exponent:  41  repeated:  01204819277108433734939759036144578313253
 prime:  89  exponent:  44  repeated:  01123595505617977528089887640449438202247191
 prime:  97  exponent:  96  repeated:  010309278350515463917525773195876288659793814432989690721649484536082474226804123711340206185567
 prime:  101  exponent:  4  repeated:  0099
 prime:  103  exponent:  34  repeated:  0097087378640776699029126213592233
 prime:  107  exponent:  53  repeated:  00934579439252336448598130841121495327102803738317757
 prime:  109  exponent:  108  repeated:  009174311926605504587155963302752293577981651376146788990825688073394495412844036697247706422018348623853211
 prime:  113  exponent:  112  repeated:  0088495575221238938053097345132743362831858407079646017699115044247787610619469026548672566371681415929203539823

==========================================

int main()
{
          
mpz_class  p,m,n,w,x,y,z, bound, quotient, zerocount;




for( p = 3; p <= 125; p += 2) {
if( mp_PrimeQ(p) ){

   n = 1;
  mpz_class ten = 10;
  
while (p != 5 &&  ten % p != 1 && n <= p ) {  ten *= 10; n++ ; } 

zerocount = 0;
quotient = ( ten - 1 ) / p;

while ( ten > quotient )  { ten /= 10; zerocount++ ;}

if(p!=5) {
cout << " prime:  "  << p << "  exponent:  " <<  n <<   "  repeated:  ";
for(w = 1; w < zerocount; ++w) cout << "0"; 
cout << quotient << endl;
}
}}


cout << endl << endl;
  return 0;
}
 
//        


//   g++  -o mse mse.cc  -lgmp -lgmpxx