Repeating digits under long division

[CraigD]

In “Is Pi = 3.1415 or 3.33 ?”, invisible resident mathemagician Turtle notes:

Originally Posted by Turtle

___Here's a twister for you; rewrite 22/7 in base twelve notation & do the division longhand. You end up with a repeating decimal!

Any rational number in any base system has a repeating part (when the repeating sequence of digits consists of exactly one zero, we call it a terminating, but this is just a convention – those zeros do repeat forever). When the remainder of the numerator divided by the denominator is one, the repeating sequence always begins in the first place after the point.

It’d be no fun to leave this bit of textbook Math as is, so consider this:

Here is 22/7 long divided by the first 99 bases ([]s surround the repeating portion, ()s surround numerals too big to be represented in base 10):

3.[001] base 2, 3 repeating digits
3.[010212] base 3, 6 repeating digits
3.[021] base 4, 3 repeating digits
3.[032412] base 5, 6 repeating digits
3.[05] base 6, 2 repeating digits
3.1[0] base 7, 1 repeating digits
3.[1] base 8, 1 repeating digits
3.[125] base 9, 3 repeating digits
3.[142857] base 10, 6 repeating digits
3.[163] base 11, 3 repeating digits
3.[186(10)35] base 12, 6 repeating digits
3.[1(11)] base 13, 2 repeating digits
3.2[0] base 14, 1 repeating digits
3.[2] base 15, 1 repeating digits
3.[249] base 16, 3 repeating digits
3.[274(14)9(12)] base 17, 6 repeating digits
3.[2(10)5] base 18, 3 repeating digits
3.[2(13)(10)(16)58] base 19, 6 repeating digits
3.[2(17)] base 20, 2 repeating digits
3.3[0] base 21, 1 repeating digits
3.[3] base 22, 1 repeating digits
3.[36(13)] base 23, 3 repeating digits
3.[3(10)6(20)(13)(17)] base 24, 6 repeating digits
3.[3(14)7] base 25, 3 repeating digits
3.[3(18)(14)(22)7(11)] base 26, 6 repeating digits
3.[3(23)] base 27, 2 repeating digits
3.4[0] base 28, 1 repeating digits
3.[4] base 29, 1 repeating digits
3.[48(17)] base 30, 3 repeating digits
3.[4(13)8(26)(17)(22)] base 31, 6 repeating digits
3.[4(18)9] base 32, 3 repeating digits
3.[4(23)(18)(28)9(14)] base 33, 6 repeating digits
3.[4(29)] base 34, 2 repeating digits
3.5[0] base 35, 1 repeating digits
3.[5] base 36, 1 repeating digits
3.[5(10)(21)] base 37, 3 repeating digits
3.[5(16)(10)(32)(21)(27)] base 38, 6 repeating digits
3.[5(22)(11)] base 39, 3 repeating digits
3.[5(28)(22)(34)(11)(17)] base 40, 6 repeating digits
3.[5(35)] base 41, 2 repeating digits
3.6[0] base 42, 1 repeating digits
3.[6] base 43, 1 repeating digits
3.[6(12)(25)] base 44, 3 repeating digits
3.[6(19)(12)(38)(25)(32)] base 45, 6 repeating digits
3.[6(26)(13)] base 46, 3 repeating digits
3.[6(33)(26)(40)(13)(20)] base 47, 6 repeating digits
3.[6(41)] base 48, 2 repeating digits
3.7[0] base 49, 1 repeating digits
3.[7] base 50, 1 repeating digits
3.[7(14)(29)] base 51, 3 repeating digits
3.[7(22)(14)(44)(29)(37)] base 52, 6 repeating digits
3.[7(30)(15)] base 53, 3 repeating digits
3.[7(38)(30)(46)(15)(23)] base 54, 6 repeating digits
3.[7(47)] base 55, 2 repeating digits
3.8[0] base 56, 1 repeating digits
3.[8] base 57, 1 repeating digits
3.[8(16)(33)] base 58, 3 repeating digits
3.[8(25)(16)(50)(33)(42)] base 59, 6 repeating digits
3.[8(34)(17)] base 60, 3 repeating digits
3.[8(43)(34)(52)(17)(26)] base 61, 6 repeating digits
3.[8(53)] base 62, 2 repeating digits
3.9[0] base 63, 1 repeating digits
3.[9] base 64, 1 repeating digits
3.[9(18)(37)] base 65, 3 repeating digits
3.[9(28)(18)(56)(37)(47)] base 66, 6 repeating digits
3.[9(38)(19)] base 67, 3 repeating digits
3.[9(48)(38)(58)(19)(29)] base 68, 6 repeating digits
3.[9(59)] base 69, 2 repeating digits
3.(10)[0] base 70, 1 repeating digits
3.[(10)] base 71, 1 repeating digits
3.[(10)(20)(41)] base 72, 3 repeating digits
3.[(10)(31)(20)(62)(41)(52)] base 73, 6 repeating digits
3.[(10)(42)(21)] base 74, 3 repeating digits
3.[(10)(53)(42)(64)(21)(32)] base 75, 6 repeating digits
3.[(10)(65)] base 76, 2 repeating digits
3.(11)[0] base 77, 1 repeating digits
3.[(11)] base 78, 1 repeating digits
3.[(11)(22)(45)] base 79, 3 repeating digits
3.[(11)(34)(22)(68)(45)(57)] base 80, 6 repeating digits
3.[(11)(46)(23)] base 81, 3 repeating digits
3.[(11)(58)(46)(70)(23)(35)] base 82, 6 repeating digits
3.[(11)(71)] base 83, 2 repeating digits
3.(12)[0] base 84, 1 repeating digits
3.[(12)] base 85, 1 repeating digits
3.[(12)(24)(49)] base 86, 3 repeating digits
3.[(12)(37)(24)(74)(49)(62)] base 87, 6 repeating digits
3.[(12)(50)(25)] base 88, 3 repeating digits
3.[(12)(63)(50)(76)(25)(38)] base 89, 6 repeating digits
3.[(12)(77)] base 90, 2 repeating digits
3.(13)[0] base 91, 1 repeating digits
3.[(13)] base 92, 1 repeating digits
3.[(13)(26)(53)] base 93, 3 repeating digits
3.[(13)(40)(26)(80)(53)(67)] base 94, 6 repeating digits
3.[(13)(54)(27)] base 95, 3 repeating digits
3.[(13)(68)(54)(82)(27)(41)] base 96, 6 repeating digits
3.[(13)(83)] base 97, 2 repeating digits
3.(14)[0] base 98, 1 repeating digits
3.[(14)] base 99, 1 repeating digits
3.[(14)(28)(57)] base 100, 3 repeating digits

If you inspect this list, you’ll notice something peculiar – the number of repeating digits is either 1, 2, 3 or 6, depending on the base. This remains true even for dividing in the first million bases! Odder still, the number of bases for which 22/7 (or 1/7 – the part before the decimal isn’t critical to this problem) repeats in 1, 2, 3 or 6 digits appears to have the same ratio: 2 each for 1, 3, or 6 digits to 1 for 2 digits.

Such tantalizing patterns seemingly spring up everywhere numbers are examined. Is a particular one useful (other than for winning bets in bars)? I’ve no guess.

The bug in the output listing has been fixed.

[Turtle]

Originally Posted by CraigD

If you inspect this list, you’ll notice something peculiar – the number of repeating digits is either 1, 2, 3 or 6, depending on the base. This remains true even for dividing in the first million bases! Odder still, the number of bases for which 22/7 (or 1/7 – the part before the decimal isn’t critical to this problem) repeats in 1, 2, 3 or 6 digits appears to have the same ratio: 2 each for 1, 3, or 6 digits to 1 for 2 digits.

Such tantalizing patterns seemingly spring up everywhere numbers are examined. Is a particular one useful (other than for winning bets in bars)? I’ve no guess.

Ohhhhh.....what a beautiful list! Curiouser still after reading the list is that those digit/values/resultants not only appear in regimented ratios, but ordered regimental ratios! 3 6 3 2 1 1 & repeat 3 6 3 2 1 1, step 3 6 3 2 1 1, dance 3 6 3 2 1 1 Oh that perfect Six is yet again popping up in some inexhorably naturally beautifal way. A Tao Te if ever I saw one!
___Deep & long genuine genuflections in your specific direction yet again Craig for following through. I often quote myself saying "follow-through is the graceful extention of intention". Your grace is beaming off my face.

[Turtle]

___Honorable Craig; belated Happy New Year to you & yours. I may honestly say without reservation that you have contributed to the forwarding of my work in mathematics over the last year here at Hypography more than any single individual alive or dead in my entire lifetime. I am in your debt.

___I have returned to the list you proffered in this thread a number of times to read it, & it is constantly hanging 'round in my head like an ephemeral smoke ring since I first viewed it.
___Today, it bumped up with another such apparition floating 'round my caranium, the Fibonacci Sequence. Below is a list of the first few elements, in both base ten & base twelve notation. May I impose to ask you how interesting an output you think your algorithm may give for say 21/13 or 610/377 or any such pair as you may choose?
___*************************
Base Ten Base Twelve
1___________1
1___________1
2___________2
3___________3
5___________5
8___________8
13__________11
21__________19
34__________2A
55__________47
89__________75
144_________100
233_________175
377_________275
610_________42A
987_________6A3
1597________B11
2584________15B4
4181________2505
6765________3AB9
10946_______6402
17711_______A2BB
28657_______14701
46368_______22A00
75025_______37501
121393______5A301
196418______95802
.
.
.

[CraigD]

Happy New Year to you, Turtle, and all my fellow Math enthusiasts here at hypography! I’ve been an infrequent visitor for the past weeks, due to the demands of holiday cheer – my apologies in taking so long to respond to your post.

Originally Posted by Turtle

…___Today, it bumped up with another such apparition floating 'round my caranium, the Fibonacci Sequence. Below is a list of the first few elements, in both base ten & base twelve notation. May I impose to ask you how interesting an output you think your algorithm may give for say 21/13 or 610/377 or any such pair as you may choose?

Finding the repeating sequences of the 2-Fibonacci sequence is an interesting exercise in “sneaking up on an irrational number” – in this case, the famous “golden ratio” X, where X= 1 + 1/X. The number of repeating digits should approach infinity (and exceed the storage of my calculator well before that! ) as the Fibonacci ratio approaches X, but maybe it’ll do something interesting before that

Here’re the ratios of F(2,n)/F(2,n-1) for n=2 to 29, in base 10 (when there were more than about 1000 numerals, I shortened them with an “…”, so as not to needlessly bedevil Hypography’s engine or our browsers):

n F(2,n) (# of non-repeating digits, # of repeating digits) F(2,n)/F(2,n-1)
2 1 (0,1) 1.[0]
3 2 (0,1) 2.[0] 
4 3 (1,1) 1.5[0]
5 5 (0,1) 1.[6]
6 8 (1,1) 1.6[0]
7 13 (3,1) 1.625[0]
8 21 (0,6) 1.[615384]
9 34 (0,6) 1.[619047]
10 55 (1,16) 1.6[1764705882352941]
11 89 (1,2) 1.6[18]
12 144 (0,44) 1.[61797752808988764044943820224719101123595505]
13 233 (4,1) 1.6180[5]
14 377 (0,232) 1.[6180257510729613733905579399141630901287553648068669527896995708154506437768240343347639484978540772532188841201716738197424892703862660944206008583690987124463519313304721030042918454935622317596566523605150214592274678111587982832] 
15 610 (0,84) 1.[618037135278514588859416445623342175066312997347480106100795755968169761273209549071] 
16 987 (1,60) 1.6[180327868852459016393442622950819672131147540983606557377049] 
17 1597 (0,138) 1.[618034447821681864235055724417426545086119554204660587639311043566362715298885511651469098277608915906788247213779128672745694022289766970] 
18 2584 (0,133) 1.[6180338134001252348152786474639949906073888541014402003757044458359423919849718221665623043206011271133375078271759549154664996869129] 
19 4181 (3,144) 1.618[034055727554179566563467492260061919504643962848297213622291021671826625386996904024767801857585139318885448916408668730650154798761609907120743]
20 6765 (0,336) 1.[618033963166706529538387945467591485290600334848122458741927768476441042812724228653432193255202104759626883520688830423343697679980865821573786175556087060511839272901219803874671131308299449892370246352547237502989715379095910069361396795025113609184405644582635733078210954317149007414494140157856972016264051662281750777325998564936]
21 10946 (1,10) 1.6[1803399852]
22 17711 (1,420) 1.6[180339850173579389731408733784030696144710396491869175954686643522748035812168828795907180705280467750776539375114196966928558377489493879042572629270966563128083318107071076192216334734149460990316097204458249588890919057189841037822035446738534624520372738900054814544125708021194957061940434862050063950301479992691394116572263840672391741275351726658139959802667641147451123698154577014434496619769778914672026310981]
23 28657 (0,396) 1.[618033990175597086556377392580881937778781548190390153012252272598949805205804302410931059793348766303427248602563378691208853255039241149568065044322737281915193947264411947377336118796228332674609000056462085709446106939190333690926542826492010614872113375868104567782733894190051380497995595957314663203658743153972107729659533623172039975156682287843712946756253175992321156343515329456270114]
24 46368 (0,28656) 1.[618033988205325051470844819764804410789684893743238999197403775691803049865652371148410510520989636040060020239383047771923090344418466692256691209826569424573402659036186621069895662490839934396482534808249293366367728652685207802631119796210350001744774400669993369857277454025194542345674704260739086436123809191471542729525072408137627804724849077014342045573507345500226820672087099138081446069023275290504937711553896081236696095194891300554838259413057891614614230380011864465924555954915029486 … 0177]
25 75025 (5,66) 1.61803[398895790200138026224982746721877156659765355417529330572808833678]
26 121393 (2,1500) 1.61[8033988670443185604798400533155614795068310563145618127290903032322559146951016327890703098967010996334555148283905364878373875374875041652782405864711762745751416194601799400199933355548150616461179606797734088637120959680106631122959013662112629123625458180606464511829390203265578140619793402199266911029656781072975674775074975008330556481172942352549150283238920359880039986671109630123292235921359546817727424191936021326224591802732422525824725091636121292902365878040653115628123958680439853 … 0589]
27 196418 (0,3016) 1.[618033988780242682856507376866870412626757720791149407296961109783924938011252708146268730486930877398202532271218274529832856919262230935885924229568426515532197078908998047663374329656569983442208364567973441631725058281779015264471592266440404306673366668588798365638875388201955631708582867216396332572718361025759310668654700023889351115797451253367162851235244206832354419118071058463008575453279843154053363867768322720420452579638035142059262066181740298040249437776478050628948950927977725239 … 8575]
28 317811 (1,5616) 1.6[18033988738303006852732437963934058996629636794998421733242370862140944312639371137064831125456933682249081041452412711665936930423891904000651671435408160148255251555356433728069728843588673135863311916423138408903460986263988025537374375057275809752670325530246718732499058131128511643535724831736398904377399220030750745858322556995794682768381716543290329806840513598550031056216843670132065289331934955044853323015202272704 … 1121]
29 514229 (0,84) 1.[618033988754322537608830405492572629644663022991652271318488032195235533068395996362]

Here they are again, in base 12:

2 1 (0,1) 1.[0]
3 2 (0,1) 2.[0] 
4 3 (1,1) 1.6[0]
5 5 (1,1) 1.8[0]
6 8 (0,4) 1.[7249]
7 13 (2,1) 1.76[0]
8 21 (0,2) 1.[74]
9 34 (1,6) 1.7[5186a3]
10 55 (1,16) 1.7[4b36429a70857921]
11 89 (0,4) 1.[7502]
12 144 (0,8) 1.[74ba4701]
13 233 (2,1) 1.75[0]
14 377 (0,8) 1.[74bb4700]
15 610 (0,4) 1.[74bb]
16 987 (1,60) 1.7[4bb6401a7b45426271b1743566a198749164b5a4206739968899108b7901] 
17 1597 (1,138) 1.7[4bb68b808906694b1083962a181731253a9bb15b415611169a214770583432624a797ba2ba82b0223178429320b468650499373b85b945a0446334856641a9150a0976727b] 
18 2584 (0,266) 1.[74bb670b330627b0560065aa584217367971901429222a5402a75ba46346ab5721407268b7b3855347bb341604a51a3317257ba24391020a781a0021b75a948652272470054b088b5940b65bb56116379a485424a2ba792999167b9146017587510649a7b495304083668740087a5b716a188a4964019782ab9b143a1bb9a0461273569949]
19 4181 (2,48) 1.74[bb67975739b04aa21a35aa7608036970315478892691808a]
20 6765 (0,1008)  1.[74bb67638024130246264642270965978328669762517979118aa56a760440aa72882782990479662725466301b684284600753328b6398049a411443972596051bb2aa80786b42221343b80083237075828b1435a6a21680112858a71a92346a4739101098b6892267a037987021264b3848a928410653536650b6556115229b618a56085808165640195a93b276bb25b1a5a731789a0952b119895b235494b750461855ba67845711315308ba5499b172259abb41927125bb962ab101a893a48677666897209ba340304539518273336a6813b92b297 … 9b74]
21 10946 (1,40) 1.7[4bb6776a53584854a68400309478b88b03926563]
22 17711 (1,30) 1.7[4bb67710916859a700544ab2a53621]
23 28657 (0,264) 1.[74bb677334621038707a41b3059452b67a951b1b729ab4931b193177758790673a3783a4aab726497433427b2a88589032415958573563320aa1a16ab3385009a301000120717b638536b77bb6b1509607b505969419b9a06991661aa67600894565853938a335b89b47bb34b779758a118a79431256a237183055a356693a3a05082a02]
24 46368 (0,7164) 1.[74bb67725262b37b6869444b969083391096ba8470a6b1b5655622536667b157423940307b40005956b451666945a42371b7b494b06b33469694383724006bb2a8a244b23b100941a71334542475702683856423a08550bb83a3b638a86b36337056a56a702b911058820977a26785767b95162b44790ba0089789959725990aba5b78ba103897678a8406897458370a77a19b6126035734226a4382a6177642b0a81774aa6406a2b9264a5634233668136255584b8721646a9b855187a83472768a781068b950968bb714801a80a2641078686b6501088 … 9201]
25 75025 (3,66) 1.74(11)[b67729114bb13256229ba264b0457b8509a08b3b4a17815a7a983342b939746685] 
26 121393 (0,300) 1.[74bb67727739b9511071b253051695011b9261701a94207b04624107b079bb3250557b94700bbbb88249a2a72201a74b55800847319146622b29b186012298386762243ab22a02190029bb470054494482026aab4a0968b6a526baa0295a4ba1279b40b7597ab40b4200896b6640274bb000033972191499ba1470663bb3748a2a755990920a35ba992383545997810991b9a2bb9200]
27 196418 (0,40) 1.[74bb67728214b70392004700544939a704b829bb] 
28 317811 (1,5616) 1.7[4bb67727b6303a9a208a9767279246b4399b53467152459003a2a68733663395b617017b281b98b320872681361048ba79656921519b2571a8563a43041939bb87a72011832739424919044a0843b19573a6927b8444516b12544646998a2b1319885989a85212324a7848a591230b19893b429964825a972201a41969577a85507b1785b537b170a8a7412798870b30109b096558611bb6829a626a2a1024337a1415359458a67033195901970a216499b0005341393a71a4a55451421b443b5200b6b210071b270070801a7b4a1767440525168ba4a … 3601] 
29 514229 (1,280) 1.7[4bb67728061a380b0474b12b14abb7aa52637273a97a68b82072a60a0104453739744b54a0b40a307a36979a1758b3363603a3610578619708873356a815221830a9b0893a780000948b459913a847425a8837075408b95399436238a2bb2b4465a94ab30680124300626b03418691807419345a588115b36856463aaa204330186063a2299b1b095b2a113b]

Other than a bit of surprise at how quickly and un-monotonically the # of repeating digits gets large, and that they don’t get as large as fast in base 12 as in base 10, nothing remarkable jumps out at me in this, but then, I haven’t looked at it much.

[Turtle]

Originally Posted by CraigD

Here’re the ratios of F(2,n)/F(2,n-1) for n=2 to 29, in base 10 (when there were more than about 1000 numerals, I shortened them with an “…”, so as not to needlessly bedevil Hypography’s engine or our browsers):

n F(2,n) (# of non-repeating digits, # of repeating digits) F(2,n)/F(2,n-1)
2 1 (0,1) 1.[0]
3 2 (0,1) 2.[0] 
4 3 (1,1) 1.5[0]
5 5 (0,1) 1.[6]
6 8 (1,1) 1.6[0]
7 13 (3,1) 1.625[0]
8 21 (0,6) 1.[615384]
9 34 (0,6) 1.[619047]
10 55 (1,16) 1.6[1764705882352941]
11 89 (1,2) 1.6[18]
12 144 (0,44) 1.[61797752808988764044943820224719101123595505]
13 233 (4,1) 1.6180[5]
14 377 (0,232) 1.[6180257510729613733905579399141630901287553648068669527896995708154506437768240343347639484978540772532188841201716738197424892703862660944206008583690987124463519313304721030042918454935622317596566523605150214592274678111587982832] 
15 610 (0,84) 1.[618037135278514588859416445623342175066312997347480106100795755968169761273209549071] 
16 987 (1,60) 1.6[180327868852459016393442622950819672131147540983606557377049] 
17 1597 (0,138) 1.[618034447821681864235055724417426545086119554204660587639311043566362715298885511651469098277608915906788247213779128672745694022289766970] 
18 2584 (0,133) 1.[6180338134001252348152786474639949906073888541014402003757044458359423919849718221665623043206011271133375078271759549154664996869129] 
19 4181 (3,144) 1.618[034055727554179566563467492260061919504643962848297213622291021671826625386996904024767801857585139318885448916408668730650154798761609907120743]
20 6765 (0,336) 1.[618033963166706529538387945467591485290600334848122458741927768476441042812724228653432193255202104759626883520688830423343697679980865821573786175556087060511839272901219803874671131308299449892370246352547237502989715379095910069361396795025113609184405644582635733078210954317149007414494140157856972016264051662281750777325998564936]
21 10946 (1,10) 1.6[1803399852]
22 17711 (1,420) 1.6[180339850173579389731408733784030696144710396491869175954686643522748035812168828795907180705280467750776539375114196966928558377489493879042572629270966563128083318107071076192216334734149460990316097204458249588890919057189841037822035446738534624520372738900054814544125708021194957061940434862050063950301479992691394116572263840672391741275351726658139959802667641147451123698154577014434496619769778914672026310981]
23 28657 (0,396) 1.[618033990175597086556377392580881937778781548190390153012252272598949805205804302410931059793348766303427248602563378691208853255039241149568065044322737281915193947264411947377336118796228332674609000056462085709446106939190333690926542826492010614872113375868104567782733894190051380497995595957314663203658743153972107729659533623172039975156682287843712946756253175992321156343515329456270114]
24 46368 (0,28656) 1.[618033988205325051470844819764804410789684893743238999197403775691803049865652371148410510520989636040060020239383047771923090344418466692256691209826569424573402659036186621069895662490839934396482534808249293366367728652685207802631119796210350001744774400669993369857277454025194542345674704260739086436123809191471542729525072408137627804724849077014342045573507345500226820672087099138081446069023275290504937711553896081236696095194891300554838259413057891614614230380011864465924555954915029486 … 0177]
25 75025 (5,66) 1.61803[398895790200138026224982746721877156659765355417529330572808833678]
26 121393 (2,1500) 1.61[8033988670443185604798400533155614795068310563145618127290903032322559146951016327890703098967010996334555148283905364878373875374875041652782405864711762745751416194601799400199933355548150616461179606797734088637120959680106631122959013662112629123625458180606464511829390203265578140619793402199266911029656781072975674775074975008330556481172942352549150283238920359880039986671109630123292235921359546817727424191936021326224591802732422525824725091636121292902365878040653115628123958680439853 … 0589]
27 196418 (0,3016) 1.[618033988780242682856507376866870412626757720791149407296961109783924938011252708146268730486930877398202532271218274529832856919262230935885924229568426515532197078908998047663374329656569983442208364567973441631725058281779015264471592266440404306673366668588798365638875388201955631708582867216396332572718361025759310668654700023889351115797451253367162851235244206832354419118071058463008575453279843154053363867768322720420452579638035142059262066181740298040249437776478050628948950927977725239 … 8575]
28 317811 (1,5616) 1.6[18033988738303006852732437963934058996629636794998421733242370862140944312639371137064831125456933682249081041452412711665936930423891904000651671435408160148255251555356433728069728843588673135863311916423138408903460986263988025537374375057275809752670325530246718732499058131128511643535724831736398904377399220030750745858322556995794682768381716543290329806840513598550031056216843670132065289331934955044853323015202272704 … 1121]
29 514229 (0,84) 1.[618033988754322537608830405492572629644663022991652271318488032195235533068395996362]

Other than a bit of surprise at how quickly and un-monotonically the # of repeating digits gets large, and that they don’t get as large as fast in base 12 as in base 10, nothing remarkable jumps out at me in this, but then, I haven’t looked at it much.

___After a first few scans (looking overs? ) of the lists (thank you for the lists Craig!), I see no great "surprises" yet either. I did notice that in the base ten list that 15 & 29 share the same output; I have marked them in red above.
___Just to clarify, is "a" ten, "b" eleven etc..? Some sources use uppercase "A" for ten, etc..
___

[CraigD]

Originally Posted by Turtle

… I did notice that in the base ten list that 15 & 29 share the same output; I have marked them in red above.

610/377 and 514229/317811 share the same number of repeating digits, but don’t have the same value. I suspect this is a coincidence of no great significance – since the maximum number of repeating digits for a particular divisor is usually less than that divisor, division by a large collection of divisors is likely to produce many similar coincidences. The computational demand of finding the number of repeating digits as this Rational number from this sequence approaches the Irrational number “the golden ratio”, (5^.5+1)/2, and the number of repeating digits approaches infinity, makes number-spelunking a CPU-intense task, even in this CPU-rich day and time!

Just to clarify, is "a" ten, "b" eleven etc..? Some sources use uppercase "A" for ten, etc.

Yes. In my previous list, I showed such “digits” as “(10)”, “(11)”, etc, but changed it to the “a”, “b”, etc. convention in this last one.

Here’s something I find interesting – the number of repeating digits for the ratio of various consecutive 2-Fibonachi sequence terms appears to “clump up” into just a few values for many number bases. Here’s a list for F(2,2)/F(2,1) to F(2,29)/F(2,28) in bases 2-16

F(2,n)/F(2,n-1)  #repeating digits:base,base … ……
1/1  1:2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
2/1  1:2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
3/2  1:2,3,4,5,6,7,8,9,10,11,12,13,14,15,16
5/3  1:3,4,6,7,9,10,12,13,15,16 2:2,5,8,11,14
8/5  1:5,6,10,11,15,16 2:4,9,14 4:2,3,7,8,12,13
13/8  1:2,4,6,8,9,10,12,14,16 2:3,5,7,11,13,15
21/13  1:13,14 2:12 3:3,9,16 4:5,8 6:4,10 12:2,6,7,11,15
34/21  1:7,15 2:6,8,13,14 3:4,9,16 6:2,3,5,10,11,12
55/34  2:16 4:4,13 8:2,8,9,15 16:3,5,6,7,10,11,12,14
89/55  1:11 2:10 4:12 5:5,15,16 10:4,6,9,14 20:2,3,7,8,13
144/89  8:12 11:2,4,8,16 22:11 44:5,9,10 88:3,6,7,13,14,15
233/144  1:6,10,12 2:8,9,15 3:4,16 4:3 6:2,7,14 12:5,11,13
377/233  8:12 29:2,4,8,16 116:7,9,13,14,15 232:3,5,6,10,11
610/377  4:12 14:13 21:16 28:5,8,14 42:4,9 84:2,3,6,7,10,11,15
987/610  4:11 6:14 10:9 12:13 15:15,16 20:3,8 30:4,5 60:2,6,7,10,12
1597/987  23:7 46:6,8,13,14,15 69:4,9,16 138:2,3,5,10,11,12
2584/1597  19:3,9 84:14 133:10,16 266:4,12 532:2,5,6,8,15 798:7 1596:11,13
4181/2584  18:16 24:8 36:4,13 48:7,11,12 72:2,9,15 144:3,5,6,10,14
6765/4181  36:15 63:16 84:8,14 112:6 126:4,7 168:11 252:2 336:10 504:9,13 1008:3,5,12
10946/6765  5:16 10:4,10 20:2,5,8,9 40:3,6,7,11,12,13,14,15
17711/10946  20:13 30:12 84:6 105:3,9,16 140:8 210:4 420:2,5,7,10,11,14,15
28657/17711  22:11 33:8 99:2,4,16 132:5 264:12 396:9,10 792:3,6,7,13,14,15
46368/28657  597:8 796:6 1791:2,4,16 3184:11 3582:9 7164:3,12 9552:7,15 28656:5,10,13,14
75025/46368  22:6,8,15 33:4,16 66:2,10,12,14 132:7,9 264:3,5,11,13
121393/75025  25:11 250:5,9 300:12 375:16 500:3,8,15 600:13 750:4,6 1000:7 1500:2,10 3000:14
196418/121393  40:12 1160:5 1885:16 3016:10 3770:4 7540:2,8,9,13 15080:3,6,7,11,14,15
317811/196418  234:16 312:8 468:4 936:2 1404:13 2808:9,15 5616:3,5,6,7,10,11,12,14
514229/317811  84:10 105:16 140:5,8,14 168:6 210:4 280:12,13 420:2,7,9 840:3,11,15

Though I’m not finding much of note about the Fibonachi sequence, I’m encountering a lot of interesting questions about Rational numbers and repeating digits in general.

PS: I haven’t been posting the code that generated these data lately, since I’m pretty sure I’m the only practitioner of my obscure programming language (M, also known as MUMPS) haunting Hypography. If anyone wants to see it, though, let me know.

[Turtle]

Originally Posted by CraigD

610/377 and 514229/317811 share the same number of repeating digits, but don’t have the same value. I suspect this is a coincidence of no great significance – since the maximum number of repeating digits for a particular divisor is usually less than that divisor, division by a large collection of divisors is likely to produce many similar coincidences. The computational demand of finding the number of repeating digits as this Rational number from this sequence approaches the Irrational number “the golden ratio”, (5^.5+1)/2, and the number of repeating digits approaches infinity, makes number-spelunking a CPU-intense task, even in this CPU-rich day and time!

Here’s something I find interesting – the number of repeating digits for the ratio of various consecutive 2-Fibonachi sequence terms appears to “clump up” into just a few values for many number bases. Here’s a list for F(2,2)/F(2,1) to F(2,29)/F(2,28) in bases 2-16[code]F(2,n)/F(2,n-1)

Though I’m not finding much of note about the Fibonachi sequence, I’m encountering a lot of interesting questions about Rational numbers and repeating digits in general.

PS: I haven’t been posting the code that generated these data lately, since I’m pretty sure I’m the only practitioner of my obscure programming language (M, also known as MUMPS) haunting Hypography. If anyone wants to see it, though, let me know.

___I'll spare reposting the whole list, but I found this line interesting:
377/233 8:12 29:2,4,8,16 116:7,9,13,14,15 232:3,5,6,10,11
It is the only occurence of 29 repeating digits in as far as the list extends.

___Ahh the computational intensity! I liken it to if youv'e got it, use it, and if you don't got it, see if you can't borrow it and use it. My specific limitation in using MUMPS is that I don't own this machine & I am not allowed to put software on it. My specific non-limitation is I have no end of proposals for such experimental investigations such as this using MUMPS. Very deep genuflection again Craig.
___How may we encourage a wider spread distributed-processing venue do you(all) think? What level of students in a class for example might use MUMPS in a project that extends Craigs list(s) above? They already hint at number mysteries worth plumbing as far as they go, so as projects they may warm us twice.

[CraigD]

Here’s a summary of the “interesting questions”, and some observations, about representing Rational numbers as repeating fractions. I’ve presented them is “semi-formal” language that’s endeavors to sound natural at the expense of being a bit vague about distinguishing classes/sets from instances/members, but hopefully anyone with an inclination toward Analysis will be able to make sense of it.

• Definition 1: Q = 1/N = F/(B^Lf) +Sum(J>0)(R/B^(Lf+Lr*J)), where: B in _N_, >1, N,Lr in _N_, >0; F, R, Lr in _Z_, >=0
• Theorem 1: Lf+Lr <= N
• Theorem 2: Lr < N
• Definition 2: N =instance= Nrm is *maximally repeating* for B (LrMax(B)) if Lr>1 & Lr = Nm-1
• Definition 3: N =instance= Nfp is *non-repeating digits prime* for B (LfPrime(B)) if, for all N<Nfp, Lf(N)<Lf(Nfp)
• Definition 4: N =instance= Nrmp(B) is *prime maximally repeating* for B (LrMPrime(B)) if there are no Nrm<Nrmp
• Theorem 5: Nrmp(B) is unique
• Proposition 1: All Nrm are prime.
• Theorem 4: Not all primes are Nrm
• Definition 5: NfpZU(B,J) = Z(B)*U(B)^J, where Z(B), U(B) in _N_, J in _Z_, >=0
• Proposition 2: There's no Nfp >= Z(B) not in {NfpZU}
• Proposition 3a: Lr(NfpZU(B,J)) = Lr(Nrmp(B))
• Proposition 3b: R(NfpZU(B,J)) mod R(Nrmp(B)) = 0

By “theorem”, I mean I’m aware of a proof, while by “proposition”, I mean I’m not, but suspect the statement is true.

The implied challenge, then, is to prove the proposition, but mathematically credulous sorts (like me) the kind of which our computer age is so packed can leap straight to some more computational challenges, such as:

• Challenge 1: Write an efficient algorithm to generate all Nrm(B)
• Challenge 2: Write an efficient algorithm to generate all Z(B), U(B)

Here are some examples for B=10 and 2

1/N=usual notation (B,Lf,Lr.F,R)  Remarks
1/2=0.5[0] (10,1,1,5,0)
1/3=0.[3] (10,0,1,0,3)
1/4=0.25[0] (10,2,1,25,0)
1/5=0.2[0] (10,1,1,2,0)
1/6=0.1[6] (10,1,1,1,6)
1/7=0.[142857] (10,0,6,0,142857)  LrMax(10)  LrMPrime(10)  LfPrime(10)  Z(10)*U(10)^0
1/8=0.125[0] (10,3,1,125,0)
1/9=0.[1] (10,0,1,0,1)
1/10=0.1[0] (10,1,1,1,0)
1/11=0.[09] (10,0,2,0,9)
1/12=0.08[3] (10,2,1,8,3)
1/13=0.[076923] (10,0,6,0,76923)
1/14=0.0[714285] (10,1,6,0,714285)  LfPrime(10)  Z(10)*U(10)^1
1/15=0.0[6] (10,1,1,0,6)
1/16=0.0625[0] (10,4,1,625,0)
1/17=0.[0588235294117647] (10,0,16,0,588235294117647)  LrMax(10)
1/18=0.0[5] (10,1,1,0,5)
1/19=0.[052631578947368421] (10,0,18,0,52631578947368421)  LrMax(10)
…
1/120259084288=0.0000000000083153801304953438895089[285714] (10,34,6,83153801304953438895089,285714)  LfPrime(10)  Z(10)*U(10)^34

1/2=0.1[0] (2,1,1,1,0)
1/3=0.[01] (2,0,2,0,1)  LrMax(2)  LrMPrime(2)  LfPrime(2)  Z(2)*U(2)^0
1/4=0.01[0] (2,2,1,1,0)
1/5=0.[0011] (2,0,4,0,3)  LrMax(2)
1/6=0.0[01] (2,1,2,0,1)  LfPrime(2)  Z(2)*U(2)^1
1/7=0.[001] (2,0,3,0,1)
1/8=0.001[0] (2,3,1,1,0)
1/9=0.[000111] (2,0,6,0,7)
1/10=0.0[0011] (2,1,4,0,3)
1/11=0.[0001011101] (2,0,10,0,93)  LrMax(2)
1/12=0.00[01] (2,2,2,0,1)  LfPrime(2)  Z(2)*U(2)^2
1/13=0.[000100111011] (2,0,12,0,59)  LrMax(2)
1/14=0.0[001] (2,1,3,0,1)
1/15=0.[0001] (2,0,4,0,1)
1/16=0.0001[0] (2,4,1,1,0)
1/17=0.[00001111] (2,0,8,0,15)
1/18=0.0[000111] (2,1,6,0,7)
1/19=0.[000011010111100101] (2,0,18,0,13797)  LrMax(2)
…
1/51539607552=0.0000000000000000000000000000000000[01] (2,34,2,0,1)  LfPrime(2)  Z(2)*U(2)^34

Here’s a listing of Nrmp for the first dozen or so Bs for B=2 to 9999

Nrmp(B): B B …
none: 4 9 16 25 36 49 64 81 100 121 144 169 196 225 256 289 324 361 400 441 ...
3: 2 5 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 ...
5: 3 7 12 13 18 22 27 28 33 37 42 43 48 52 57 58 63 67 72 73 78 82 87 88 ...
7: 10 19 24 31 40 45 54 61 66 75 94 96 115 124 129 136 145 150 159 166 171 ...
11: 6 30 39 46 51 79 84 85 90 105 106 139 151 156 160 184 189 195 204 205 ...
13: 15 76 111 141 154 175 210 214 219 240 279 280 301 319 331 340 345 366 ...
17: 91 99 109 114 126 130 165 181 190 231 235 265 286 295 300 309 330 351 ...
19: 21 34 60 70 135 174 186 421 450 489 610 781 846 984 1105 1131 1191 1275 ...
23: 120 274 291 406 429 511 559 594 595 664 681 835 939 1045 1050 1054 1056 ...
29: 55 69 246 714 715 814 1005 1071 1266 1290 1429 1651 1726 1980 2106 2410 ...
31: 1035 1386 1665 2431 2440 2925 4599 4641 4785 5490 5509 6141 6306 6315 ...
37: 616 790 1596 1794 1869 1885 1981 2755 2830 3039 3315 3591 3705 4146 4645 ...
41: 399 1126 3045 3094 3336 3381 3619 4234 5559 6061 6579 6985 7734 7930 ...
43: 2155 2211 3255 3774 6136 6154 9316 9981
47: 510 2661 2991 5109 5340 6600 7314 7821 7875 8056 8385 8551
53: 7665 8220
59: 2634

PS:

Originally Posted by Turtle

___How may we encourage a wider spread distributed-processing venue do you(all) think?

I haven’t forgotten this “meta-question”, but am at a loss for a brief answer to what seems a profound social and technical question. For the nonce, I’ve focused just on the numbers

[TheBigDog]

CraigD, I would be interested in the code that you used to determine the number of repeating digits and how you transformed the values into different bases. I have not heard of MUMPS, but I have been toying with functions in vb.net. Among them is a function that will allow me to use numbers up to 2 billion digits long, and another for looking at prime numbers. Hopefully I could follow the logic of your code and make it work in vb.net.

Thanks,

Bill

[CraigD]

Originally Posted by webenton

CraigD, I would be interested in the code that you used to determine the number of repeating digits and how you transformed the values into different bases.

Here’s the code I used, and a few examples of its use and output

USER>r X1,!,X2,!,X21,!,X3,! ;read subroutines into symbols
k R,Q f R=1:1 s Q(R)=N\D,N=N#D q:$g(R(N))  s R(N)=R,N=N*B ;X1: find repeating digits -requires N,D,B
w Q(1),"." x "f I=2:1:R(N) s J=Q(I) x X21" w "[" x "f I=R(N)+1:1:R x X21" w "]" ;X2: display find -requires R,Q
w $s(Q(I)<10:Q(I),1:"("_Q(I)_")") ;X21: called by X1 to display 1 number
w " (",R(N)-1,",",R-R(N)," base ",B,")" ;X3: display # non-repeating,repeating digits
 
USER>s B=10,N=1,D=7 x X1,X2,X3 ;find and display 1/7 base 10
0.[142857] (0,6 base 10)
USER>s B=2,N=1,D=7 x X1,X2,X3 ;find and display 1/7 base 10
0.[001] (0,3 base 2)
USER>s B=10,N=1,D=120259084288 x X1,X2,X3 ;find and display 1/120259084288 base 10
0.0000000000083153801304953438895089[285714] (34,6 base 10)
USER>S B=16,N=1,D=3584 x X1,X2,X3 ;find and display 1/3584 base 16
0.001[249] (3,3 base 16)
USER>S B=12,N=1,D=3584 x X1,X2,X3 ;find and display 1/3584 base 12
0.00059[5186(10)3] (5,6 base 12)

The text following the “USER>” prompt is entered at the MUMPS direct mode (interpreter) prompt.
Lines that don’t begin “USER>” are either input (lines 2-5) or output (lines 8, 10, 12 & 14)

A brief summary of the keywords (M[UMPS] keywords can be, and usually are, abbreviated, usually with a single letter), operators and syntactic element used in the above:
r[ead] – reads inputted data into storagerefs (variables) specified in argument list
! – as an argument in a r[ead] or w[rite] command, outputs a newline
k[ill] – removes specified storagerefs and descendents (eg: “k a” removes a, a(1), a(1,2), a(2), etc.)
f[or] – loop structuring command. Argument is control variable, begin, increment, & end value (all optional)
s[et] – assigns the value of an expression to a storageref
\ - integer division operator
# - modulo operator
q[uit] – exit loop or other flow control structure
: - following any command, cause execution to be conditional on following expression
$g[et] – function to return the value of a variable that may not be defined
; - comment – rest of line is not executed
w[rite] – output expressions
x[ecute] – invokes code specified by expression
$s[elect] – function to return expression based on conditional expressions
_ - string concatenation operator (all values in M can be considered strings)

A good source of M language documentation can be found at http://platinum.intersystems.com:197...e.cls?KEY=RCOS (although this commercial vendor refers to it as “Cache object script”). A good general description, with links, can be found at the wikipedia article “MUMPS”. Note that all keywords beginning “Z” are implementation-specific, not standard M.

The X1 xecute code subroutine is simply an implementation of long division in base B. It keeps dividing, storing the position of each encountered remainder N as R(N). Encountering the same remainder for the 2nd time indicates the end of the repeating digits. R(N)-1 gives the number of non-repeating, N-R(N) the number of repeating digits. For large divisors D with potentially many non-repeating and repeating digits, X1 may create many nodes R(N), using a lot of time and memory.

X1 consists of only 57 characters of interpreted code. One of the reasons I like M is that it is very terse. (although this terseness leads some to comment that it looks like line noise )