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

I'm sorry... did you seriously just ask if $\frac{1}{65537}$ is a rational number or not? Pray, tell, what definition do you know for rational numbers? — JMoravitz, Jul 17 '21 at 17:19
The definition of rational numbers, usually, is that they can be written as $a/b$, where $a,b$ are integers. So, because $1$ and $65537$ are integers, $1/65537$ is indeed rational, and it has a finite repeating decimal expansion, as every other rational. You can check any online reference about rationals, like this Wikipedia page (or the question linked before) for why. — AnilCh, Jul 17 '21 at 17:26
I believe it to be a rational number by definition.
But it seems to me a rational number's decimal portion should not be infinite and non-repeating, as this is characteristic of an irrational #.

If it is a rational number, which I believe it to be, than what is its repetend length? It must be over 9999 but I don't know how to figure it out.

It is strange though because most rational numbers that are repetends have a repetend length that are shorter than 9 and the largest I have found myself is 256. — DR Wu, Jul 17 '21 at 17:28
$\phi(65537)=65536$ as $65537$ is prime and $\phi(p)=p-1$. It will be some factor of $65536$. If, as you claim, you already tested substrings of length up to $9999$, then that still leaves as possibilities $16384$, $32768$, or $65536$ itself. The punchline is though that just because you didn't find it doesn't mean it doesn't exist. — JMoravitz, Jul 17 '21 at 17:29
So if it is a factor of 65536, than it must be either 16384, 32768 or 65536? I assume this to be true having eliminated all possibilities less than 10,000.
https://www.2dtx.com/prime/prime65536.html — DR Wu, Jul 17 '21 at 17:31
Thank you JMoravitz but your link above is WAY above my understanding. I am not looking for a proof as much as one particular number that seemed to me have such large repeating sections. Is there a proof to show what the largest repetend is or does of the length of the repeating portions stretch to infinity? — DR Wu, Jul 17 '21 at 17:37
"Most rational numbers that are repetends have a repetend length that are shorter than 9 and the largest I have found myself is 256" Consider the following number... $0.\underbrace{000000000\dots000}{n~\text{zeroes}}1\underbrace{000000\dots00}{n~\text{zeroes}}1\cdots$ always $n$ zeroes followed by a $1$, repeating that pattern. Choose your favorite large $n$. The numbers like this have unbounded repetend length yet are all rational of the form $\frac{1}{10^{n+1}-1}$ — JMoravitz, Jul 17 '21 at 17:38
So how do I calculate the repetend length for 1/65537? I suck at math, I have to figure out even the keyboard syntax for division. All I did was punch it in a calculator that goes down to 9999 digits and did a search for "15258" which are the first 5 digits after the repeating 0's. — DR Wu, Jul 17 '21 at 17:46
And if you can't help with that can you tell me what the largest known repetend length is for any rational number? — DR Wu, Jul 17 '21 at 17:48
What did you not understand about my previous comment? It is very well known that there is no largest repetend length... that it is unbounded... Any repetend length or $n$ you suggest as being a candidate for being largest I could use my $0.\overline{0000\dots 001}$ example with more zeroes than the $n$ you suggested... just like how there is not any largest number, any number you suggest as being largest I could just add one to it and it would be larger. — JMoravitz, Jul 17 '21 at 17:54
As for how to calculate the repetend length, as alluded to in the linked post, it is going to be some divisor of Euler's totient function of the denominator. Which divisor however will depend on the number itself. I do not personally know how to tell beyond checking, however it is not a very useful piece of information to know in the first place. I know it repeats, so why do I care how quickly it repeats. — JMoravitz, Jul 17 '21 at 17:57
See further: Length of period of decimal expansion of a fraction and the length of the repetend of $1/p$ is equal to the order of $10$ modulo $p$. — JMoravitz, Jul 17 '21 at 18:01
Following what was shown in those posts, we can see from calculations in wolframalpha that $10^{32768}\equiv -1\pmod{65537}$, which implies that $32768$ is not the repetend length and instead indeed $\frac{1}{65537}$ does have a full $65536$ length repetend. — JMoravitz, Jul 17 '21 at 18:06
I guess what the big difference here is I am not coming at this from a math perspective as much as a pattern recognition perspective. When you look at it seems like a random number generator. I searched also for "59741" which is as far as the calculator will go and it also does not repeat.
1/65536 obviously terminates. 1/65538 gives a repetend length of 110, which immediately begins following 4 zeros:

0.000015258323415423113308309682932039427507705453324788672220696389880679910891391253929018279471451676889743355000[keeps repeating from here]. — DR Wu, Jul 17 '21 at 18:07
"I am not coming at this from a math perspective" That was obvious — JMoravitz, Jul 17 '21 at 18:08

score 2 · Answer 1 · answered Jul 17 '21 at 18:08

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

Will Jagy · Answer 2 · 2021-07-17T18:44:12.977

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

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

2 Answers2