continued fraction of the roots of $x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$

Question

I would like to find the continued fraction expansion of the roots of:

$$x^2 - \frac{53793390359}{1088391168}x + \frac{823543}{12230590464} = 0$$

Eq 1.6 from [1]

What makes this problem so difficult is obtaining numerical accuracy. However, we have some starting points:

• since this is a quadratic the digits are ultimately periodic
• the continued fraction of the middle coefficient is $$ [49, 2, 2, 1, 4, 1, 1, 6, 2, 1, 6, 1, 9, 1, 454, 3, 1, 4]$$
• the constant term is of order $10^{-6}$.

Since this is a quadratic, I am hoping an exact answer is possible. Although the number $454$ may already rule out the possibility of a "small divisors" explanation of the equation here.

---

Work I have no idea what to do here. My first guess is to approximate this equation as:

$$ x^2 - 49x + 0 = x(x-49) = 0 $$

so the roots are $x = 0, 49$. If I wanted to get more accuracy I could try dividing by $x$ and hoping for the best:

\begin{eqnarray} x &=& \underbrace{\frac{53793390359}{1088391168}}_A + \underbrace{\frac{823543}{12230590464}}_{1/B}\frac{1}{x} \\\\ &=& A + \frac{1}{B\, x} = A + \frac{1}{AB + \frac{1}{x}}\end{eqnarray}

The first two digits are... $x = [\lfloor A \rfloor, \lfloor AB \rfloor,...] = [49,734015,... ]$ so $x = \mathbf{49} $ is a really good guess!

Answer 1

I computed the continued fraction out many terms, but didn't see a pattern.

Here's the Maple code that I used (following the VAS algorithm) to generate the continued fraction for the root near 49:

f := x^2-(53793390359/1088391168)*x+823543/12230590464; 
r := [evalf(solve(f = 0))]; 
m := floor(max(r)); 
cf := [m]; 
f := simplify((1+x)^2*subs(x = 1/(1+x)+49, f)); 
for a to 100 do 
    r := evalf(solve(f = 0)); 
    m := floor(max(r)); 
    cf := [op(cf), 1+m]; 
    f := simplify((1+x)^2*subs(x = 1/(1+x)+m, f)) 
end do

Here are the first 101 terms of the continued fraction:

For the smaller root: $$[0, 2, 2, 1, 4, 1, 1, 6, 1, 2, 2, 6, 1, 2, 1, 17, 1, 3, 3, 6, 4, 3, 1, 1, 2, 2, 4, 14, 35, 1, 1, 247, 1, 1, 4, 2, 15, 1, 1, 1, 16, 42, 1, 6, 3, 1, 5, 1, 4, 1, 1, 1, 4, 16, 3, 17, 1, 1, 3, 1, 1, 4, 1, 1, 11, 1, 5, 1, 71, 1, 1, 1, 1, 7, 61, 2, 1, 1, 4, 4, 62, 1, 5, 1, 1, 2, 3, 2, 3, 45, 1, 1, 3, 6, 1, 3, 1, 47, 1, 27, 1]$$

For the larger root: $$[49, 2, 2, 1, 4, 1, 1, 6, 1, 2, 2, 6, 1, 2, 1, 17, 1, 3, 3, 6, 4, 3, 1, 1, 2, 2, 4, 14, 35, 1, 1, 247, 1, 1, 4, 2, 15, 1, 1, 1, 16, 42, 1, 6, 3, 1, 5, 1, 4, 1, 1, 1, 4, 16, 3, 17, 1, 1, 3, 1, 1, 4, 1, 1, 11, 1, 5, 1, 71, 1, 1, 1, 1, 7, 61, 2, 1, 1, 4, 4, 62, 1, 5, 1, 1, 2, 3, 2, 3, 45, 1, 1, 3, 6, 1, 3, 1, 47, 1, 27, 1]$$

By the way, the continued fraction expansion does not repeat before the 10,000th term.

Answer 2

illustration only:

jagy@phobeusjunior: ./indefCycle 101 -473 -137

  0  form            101        -473        -137  delta      0
  1  form           -137         473         101


           0          -1
           1           0

To Return  
           0           1
          -1           0

0  form   -137 473 101   delta  4
1  form   101 335 -413   delta  -1
2  form   -413 491 23   delta  22
3  form   23 521 -83   delta  -6
4  form   -83 475 161   delta  3
5  form   161 491 -59   delta  -8
6  form   -59 453 313   delta  1
7  form   313 173 -199   delta  -1
8  form   -199 225 287   delta  1
9  form   287 349 -137   delta  -3
10  form   -137 473 101


  form   -137 x^2  + 473 x y  101 y^2 

Answer 3

Revised my Gauss-Lagrange method program to just print out a form every 10,000,000 steps unless something good happens. Also switched to the form with integer coefficients on page 22 or 23 of the manuscript, which is already "reduced."

Anyway, the conclusion is that the continued fraction cycle is of length greater than $4 \cdot 10^9$ and we may never know how long it is. I stopped execution this morning because my monitor would not come on, had to shut off the computer and restart to see anything on the screen.

Highlights include:

2,767,313,342  new min   form   8751 440675478108752 -156710092210752   delta  -2

because we would always like to know the "minimum," the integer of smallest absolute value represented by the original form.

gp-pari

? f =     600362847  +   440675453820928 * x      -8916100448256 * x^2
%1 = -8916100448256*x^2 + 440675453820928*x + 600362847
? 
? polroots(f) 
%2 = [-1.362369574356248045442331538 E-6 + 0.E-28*I, 
      49.42468610861698234697514403 + 0.E-28*I]~
? 
? 
? 
? x =  49.42468610861698234697514403
%3 = 49.42468610861698234697514403
? 
? contfrac(x)
%4 = [49, 2, 2, 1, 4, 1, 1, 6, 52, 23, 2, 2, 25, 3, 6, 1, 3, 2, 1, 1, 10, 2, 37]

First disagreement with my output below is the final "digit" above, it gives 37 instead of my 35



  0  form              600362847     440675453820928      -8916100448256


0  form   600362847 440675453820928 -8916100448256   delta  -49
1  form   -8916100448256 433102390108160 185540661325663   delta  2
2  form   185540661325663 309060255194492 -132958235361924   delta  -2
3  form   -132958235361924 222772686253204 271828230266951   delta  1
4  form   271828230266951 320883774280698 -83902691348177   delta  -4
5  form   -83902691348177 350337756504718 212920265818911   delta  1
6  form   212920265818911 75502775133104 -221320182033984   delta  -1
7  form   -221320182033984 367137588934864 67102858918031   delta  6
8  form   67102858918031 438096718081508 -8442794594052   delta  -52
9  form   -8442794594052 439953919699900 18815616839839   delta  23
10  form   18815616839839 425564454932694 -173921639416921   delta  -2
11  form   -173921639416921 270122102734990 174257969037543   delta  2
12  form   174257969037543 426909773415182 -17133968736729   delta  -25
13  form   -17133968736729 429788663421268 138271843961468   delta  3
14  form   138271843961468 399842400347540 -62053363347321   delta  -6
15  form   -62053363347321 344797959820312 303405165543152   delta  1
16  form   303405165543152 262012371265992 -103446157624481   delta  -3
17  form   -103446157624481 358664574480894 158426860720799   delta  2
18  form   158426860720799 275042868402302 -187067863703073   delta  -1
19  form   -187067863703073 99092859003844 246401865420028   delta  1
20  form   246401865420028 393710871836212 -39758857286889   delta  -10
21  form   -39758857286889 401466273901568 207624855093248   delta  2
22  form   207624855093248 429033146471424 -12191984717033   delta  -35
23  form   -12191984717033 424405783720886 288603703227663   delta  1
24  form   288603703227663 152801622734440 -147994065210256   delta  -2
25  form   -147994065210256 439174638106584 2230687855519   delta  197
26  form   2230687855519 439716376967902 -94632787370433   delta  -4
27  form   -94632787370433 317345921995562 246971597800199   delta  1
28  form   246971597800199 176597273604836 -165007111565796   delta  -1
29  form   -165007111565796 153416949526756 258561759839239   delta  1
30  form   258561759839239 363706570151722 -59862301253313   delta  -6
31  form   -59862301253313 354641044888034 285758335630303   delta  1
32  form   285758335630303 216875626372572 -128745010511044   delta  -2
33  form   -128745010511044 298104415671604 204529546331271   delta  1
34  form   204529546331271 110954676990938 -222319879851377   delta  -1
35  form   -222319879851377 333685082711816 93164343470832   delta  4
36  form   93164343470832 411629665054840 -66430715165329   delta  -6
37  form   -66430715165329 385538916929108 171436587848028   delta  2
38  form   171436587848028 300207434463004 -151762197631433   delta  -2
39  form   -151762197631433 306841356062728 164802666248304   delta  2
40  form   164802666248304 352369308930488 -106234244763673   delta  -3
41  form   -106234244763673 285036159651550 265802390166711   delta  1
42  form   265802390166711 246568620681872 -125468014248512   delta  -2
43  form   -125468014248512 255303436312176 257067574536407   delta  1
44  form   257067574536407 258831712760638 -123703876024281   delta  -2
45  form   -123703876024281 235983791336486 279915495960559   delta  1
46  form   279915495960559 323847200584632 -79772171400208   delta  -4
47  form   -79772171400208 314330170617032 298949555895759   delta  1
48  form   298949555895759 283568941174486 -95152786121481   delta  -3
49  form   -95152786121481 287347775554400 293281304325888   delta  1
50  form   293281304325888 299214833097376 -89219257349993   delta  -4
51  form   -89219257349993 414539225702568 62632519115504   delta  6
52  form   62632519115504 337051003683480 -321683923407257   delta  -1
53  form   -321683923407257 306316843131034 77999599391727   delta  4
54  form   77999599391727 317679952002782 -298957705663761   delta  -1
55  form   -298957705663761 280235459324740 96721845730748   delta  3
56  form   96721845730748 300095615059748 -269167472061249   delta  -1
57  form   -269167472061249 238239329062750 127649988729247   delta  2
58  form   127649988729247 272360625854238 -235046175269761   delta  -1
59  form   -235046175269761 197731724685284 164964439313724   delta  1
60  form   164964439313724 132197153942164 -267813460641321   delta  -1
61  form   -267813460641321 403429767340478 29348132614567   delta  14
62  form   29348132614567 418317945867398 -163596210952881   delta  -2
63  form   -163596210952881 236066897944126 211599180537839   delta  1
64  form   211599180537839 187131463131552 -188063928359168   delta  -1
65  form   -188063928359168 188996393586784 210666715310223   delta  1
66  form   210666715310223 232337037033662 -166393606635729   delta  -2
67  form   -166393606635729 433237389509254 9766362834631   delta  44
68  form   9766362834631 426202539938274 -321160297197289   delta  -1
69  form   -321160297197289 216118054456304 114808605575616   delta  2
70  form   114808605575616 243116367846160 -294161983807433   delta  -1
71  form   -294161983807433 345207599768706 63762989614343   delta  6
72  form   63762989614343 419948275603410 -69939956303321   delta  -6
73  form   -69939956303321 419331200036442 65614216315247   delta  6
74  form   65614216315247 368039395746522 -223815369173081   delta  -1
75  form   -223815369173081 79591342599640 209838242888688   delta  1
76  form   209838242888688 340085143177736 -93568468884033   delta  -4
77  form   -93568468884033 408462607894528 73083313455104   delta  5
78  form   73083313455104 322370526656512 -308798671979073   delta  -1
79  form   -308798671979073 295226817301634 86655168132543   delta  4
80  form   86655168132543 398014527758710 -103223251064921   delta  -4
81  form   -103223251064921 427771480760658 27141262128647   delta  15
82  form   27141262128647 386466383098752 -413011483529216   delta  -1
83  form   -413011483529216 439556583959680 596161698183   delta  738
84  form   596161698183 440378082558428 -109878500591204   delta  -4
85  form   -109878500591204 438649922171204 4052482472631   delta  108
86  form   4052482472631 436686291917092 -215914534313252   delta  -2
87  form   -215914534313252 426971845335916 13766929053807   delta  31
88  form   13766929053807 426577756000118 -222022919018121   delta  -1
89  form   -222022919018121 17468082036124 218321766035804   delta  1
90  form   218321766035804 419175450035484 -21169235018441   delta  -20
91  form   -21169235018441 427593950702156 134136759369084   delta  3
92  form   134136759369084 377226605512348 -96720252803153   delta  -4
93  form   -96720252803153 396535416912876 95519136568028   delta  4
94  form   95519136568028 367617675631348 -154555735366209   delta  -2
95  form   -154555735366209 250605265833488 212531546365888   delta  1
96  form   212531546365888 174457826898288 -192629454833809   delta  -1
97  form   -192629454833809 210801082769330 194359918430367   delta  1
98  form   194359918430367 177918754091404 -209070619172772   delta  -1
99  form   -209070619172772 240222484254140 163208053348999   delta  2
100  form   163208053348999 412609729141856 -36683374285056   delta  -11
856997  new min   form   -324518241 440675290597730 127318319881983   delta  3
1234567  form   -120259880879561 437934579168168 5006202739952   delta  87
1603662  new min   form   47941263 440675404060910 -340351986291681   delta  -1
6242824  new min   form   36554311 440675457475504 -124408157419584   delta  -3
11234567  form   -52474881577409 414756058952424 105632866180976   delta  4
12792061  new min   form   -16307076 440675473041644 68549780712519   delta  6
21234567  form   -240015607188241 382304678017922 50036340074127   delta  8
31234567  form   -265567577390864 197295639354488 146167417375503   delta  2
41234567  form   -20414066561769 427158800391196 143653840658396   delta  3
49187955  new min   form   -1792272 440675477774296 41882397784567   delta  10
51234567  form   -187925712932633 68397630764078 252116432286279   delta  1
61234567  form   -159444776862369 165763590482768 261402901309888   delta  1
71234567  form   -105287151336673 370051111202072 135954509603184   delta  2
81234567  form   -124710039158544 278061224543176 234297160850111   delta  1
91234567  form   -335500161204297 322508569971560 67200488803824   delta  5
101234567  form   -239743820434969 264851046547740 129355576224412   delta  2
111234567  form   -195977545812224 315181928303264 121002674936271   delta  3
121234567  form   -26285377265348 439635083519636 8710835128023   delta  50
131234567  form   -94246140738673 336951699664668 213956848717564   delta  1
141234567  form   -227334048333824 37811170457856 211984515532631   delta  1
151234567  form   -86921965484676 307004984099356 287449829835119   delta  1
160715429  new min   form   -262144 440675477662720 380130967686087   delta  1
161234567  form   -112679846742961 306073627889614 223007516933439   delta  1
171234567  form   -84939916820009 368174359264192 172599998880768   delta  2
181234567  form   -147950820924868 156344015420676 286837586964463   delta  1
191234567  form   -4905370733961 434512536978214 274889122206239   delta  1
201234567  form   -282926473630161 342860532007852 67722129021532   delta  5
211234567  form   -285592962687729 281144630126278 100801305535783   delta  3
221234567  form   -31799181669729 392309244933656 316740331322736   delta  1
231234567  form   -67733157525873 400556060071576 124568678711408   delta  3
241234567  form   -41768757345841 394670609600372 230016340989852   delta  1
251234567  form   -123905796234609 216099857513392 297596506936768   delta  1
261234567  form   -59245754478144 356815762928656 282203294255351   delta  1
271234567  form   -234162189321977 266334703524928 131597572463616   delta  2
281234567  form   -64953983507121 435485509110140 17501807131068   delta  25
291234567  form   -32188874491536 421085899135192 131113178295567   delta  3
301234567  form   -291072355399017 231239957531656 120865960338032   delta  2
311234567  form   -59162964411769 373298993398716 231744382168156   delta  1
321234567  form   -178679581561572 333360532348540 116221495145399   delta  3
331234567  form   -76109339460969 419207043002656 60637538834176   delta  7
341234567  form   -142503482378816 157338427610320 297255009808383   delta  1
351234567  form   -38182845947601 439128016327390 8914096464303   delta  49
361234567  form   -181625016690593 131331944117882 243560607302679   delta  1
371234567  form   -93145294060809 381268914088936 131055714230256   delta  3
381234567  form   -226516684127921 436218978493166 4313015428959   delta  101
391234567  form   -128950129919961 245123313648412 260002526172572   delta  1
401234567  form   -134637491397457 225606090025112 266078875334256   delta  1
411234567  form   -160885349613968 431891040617032 11910665055879   delta  36
421234567  form   -77674961374569 373724265120016 175491077521856   delta  2
431234567  form   -18934790450313 435441210621788 60547661144284   delta  7
441234567  form   -190891023045184 90741664881648 243543180160903   delta  1
451234567  form   -251441323061316 71812659791164 187954207967919   delta  1
461234567  form   -273929606532112 178593913031704 148121166323127   delta  2
471234567  form   -377063169236868 436563931671676 2391383506463   delta  183
481234567  form   -52376656915609 432356329570898 34666594431303   delta  12
491234567  form   -404908872154009 398884268755628 21663033069948   delta  19
501234567  form   -23741317830912 419260870807328 193915090907111   delta  2
511234567  form   -126095175334313 381902477690794 95850960231663   delta  4
521234567  form   -205510961807424 434741093279888 6319685796783   delta  69
531234567  form   -175856546727108 117126266077868 256567523594359   delta  1
541234567  form   -94968487660337 324316308513300 234324593444444   delta  1
551234567  form   -138362978954553 238126526556572 248423811414044   delta  1
561234567  form   -92392627005233 302257725411642 278255819144039   delta  1
571234567  form   -124129548476473 423629155445436 29673063040348   delta  14
581234567  form   -50010351316017 437420773312382 14286762697391   delta  30
591234567  form   -219331605782697 111844864876000 207090084628224   delta  1
601234567  form   -25157832366833 394638584860594 382140080575743   delta  1
611234567  form   -321038162819012 214526091037652 115386152206071   delta  2
621234567  form   -40388055848041 378658203237468 314528893913692   delta  1
631234567  form   -121233443693241 300919634945138 213724544892487   delta  1
641234567  form   -211356696191041 49468036723994 226805908932759   delta  1
651234567  form   -84557054742889 363506748706184 183478838180976   delta  2
661234567  form   -83754414452449 281558423213722 343025893263543   delta  1
671234567  form   -196518009372672 208235240166400 191881856332399   delta  1
681234567  form   -12107827461156 434227486813460 116481811572463   delta  3
691234567  form   -35485320380601 416760206546338 144466831857367   delta  2
701234567  form   -277144385228593 436308914120510 3454344265071   delta  126
711234567  form   -116300073184113 264044125290200 267574158604656   delta  1
721234567  form   -298782973318329 390232763093056 35069993409536   delta  11
731234567  form   -65594559685604 326394288849996 334104099631631   delta  1
741234567  form   -84124971192249 407296062861566 84115316142711   delta  5
751234567  form   -124319879196516 256210983824692 258508151733807   delta  1
761234567  form   -29908245409476 403477755583220 262474255272103   delta  1
771234567  form   -55309888843489 404156460943848 139452603929968   delta  3
781234567  form   -307156936521097 188399526585392 129168900751296   delta  2
791234567  form   -245425691002209 170646691289908 168151287554268   delta  1
801234567  form   -2893592082308 438971552299916 129497631372183   delta  3
811234567  form   -111360892857769 426687999223998 27236285599783   delta  15
821234567  form   -34043673913249 417788449343738 144282963237783   delta  2
831234567  form   -334411495718976 350183313814576 53501841783159   delta  7
841234567  form   -297957734276337 230342189457650 118420615184319   delta  2
851234567  form   -108063433198273 323582554334182 207029345852439   delta  1
861234567  form   -379797737373696 438822622282240 1072665148343   delta  409
871234567  form   -284865489112681 329772286677086 74987247681159   delta  5
881234567  form   -249878132541456 123525349161992 179023633747327   delta  1
891234567  form   -130214847116352 367767473302960 113162907106039   delta  3
901234567  form   -197527679869777 337967415571764 101217337079452   delta  3
911234567  form   -154988691853593 336856855010710 130206815863071   delta  2
921234567  form   -128549839576977 279631967402588 225595069196796   delta  1
931234567  form   -20132873456353 420467744490462 216086303983039   delta  1
941234567  form   -214764744790148 178851408731524 188819457733983   delta  1
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989484954  new min   form   252159 440675477958146 -137038755464961   delta  -3
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2761234567  form   -192428278034628 88573350422180 242102720699551   delta  1
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============================================================================
4551234567  form   -258225803190249 288483139796146 107437322000951   delta  3
4561234567  form   -266358218913033 362413611247306 58991282164143   delta  6