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Why is the periodic length of simple continued fraction expansion of any quadratic irrational i.e irrational of the form $$\dfrac{P+\sqrt{R}}{Q}$$ is less than $2R$?

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Evidently the result you quote goes back to Lagrange. There is a proof in a book I do not have, Elementary Theory of Numbers by W. Sierpinski, on page 294.

The result has been improved a good deal. See Hickerson 1973 and then Cohn 1977.

The asymptotic of Cohn is $$ \frac{7}{2 \pi^2} \sqrt R \log r + O( \sqrt R) $$

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  • $\begingroup$ both those results are of specifically for $R^{1/2}$ in other words $P=0$ and $Q=1$. I am interested in how the behaviour of different $P$ and $Q$ play the role here in the upper bound for period length. $\endgroup$ – Suraj Mar 9 '17 at 21:03
  • $\begingroup$ @Suraj sure. I suggest borrowing the Sierpinski book. The one that has everything is the book by Perron in German, see references at de.wikipedia.org/wiki/Kettenbruch . By the way, I don't expect your generalisation to change the bounds on the length of the periodic part. Meanwhile, a very good inexpensive modern treatment in English is cambridge.org/us/academic/subjects/mathematics/number-theory/… $\endgroup$ – Will Jagy Mar 9 '17 at 21:09
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As I expected, allowing general reduced forms does not much change the maximum length of continued fraction periodic section. Still $\sqrt d \log d$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./bigCycle
      discr          a           b           c         length   sqrt(d) * log(d) length/(sqrt(d)*log(d))
         5           1           1          -1              2   3.598812577768002   0.5557388601882701
        17           2           1          -2              6   11.68163787745321   0.5136266046716472
        41           2           3          -4             10   23.77846330776795   0.4205486229521485
        73           4           3          -4             18   36.65770153430481   0.4910291493086477
       193           6           5          -7             30   73.11162868318813   0.4103314416643332
       241           6           5          -9             38   85.14694576902021   0.4462872937695657
       337           6           7         -12             42   106.8425201478901   0.39310192179915
       409          10           3         -10             54   121.6198623622381   0.4440064225624921
       601          12           5         -12             66   156.8634830000024   0.4207480207487106
       769          12           7         -15             70   184.2740159127738   0.3798690751556343
      1033          16           3         -16             78   223.0609527063442   0.34968020648009
      1201          16           7         -18            106   245.7386488282813   0.4313525792764952
      1609          20           3         -20            118   296.1641899980161   0.3984276424532974
      1801          20          11         -21            130   318.1208060005595   0.4086497882184147
      2161          22           7         -24            146   356.9389573465163   0.4090335251869502
      2521          24           5         -26            170   393.2619129923604   0.4322818823375415
      3361          27          11         -30            178   470.7495953275685   0.3781203462875835
      3529          29           7         -30            198   485.2689323717215   0.4080211750467672
      4201          30          11         -34            210   540.757607508973    0.3883440511680927
      4561          32           9         -35            214   569.0039278467663   0.3760958220619359
      5209          36           5         -36            238   617.6703105346307   0.3853188277642756
      5569          30          17         -44            258   643.6447653387628   0.4008422252361667
      6841          40          11         -42            290   730.3892877579093   0.397048539540084
      7561          42          13         -44            306   776.5652918414429   0.3940428489591553
      8089          44          13         -45            330   809.2933548617549   0.4077631405442236
      9241          48           5         -48            346   877.8030966889762   0.3941658457404542
     12049          54          13         -55            378   1031.460429594734   0.3664706751266437
     12289          52          15         -58            390   1043.868831383885   0.3736101589344005
     12601          52          11         -60            394   1059.851332406444   0.3717502520899829
     13729          57           7         -60            426   1116.317487006224   0.381611866658526
     15649          62           5         -63            454   1208.197114835509   0.375766499046648
     16921          64           5         -66            474   1266.506545484928   0.374257836795081
     18481          66          23         -68            502   1335.589887969525   0.375863881960938
     19009          65          17         -72            522   1358.418153458484   0.3842704830401499
     20161          69          17         -72            530   1407.329053131947   0.3765999137305586
     21121          70          11         -75            542   1447.206042048553   0.3745147437560353
     21961          70          11         -78            566   1481.483319630914   0.3820495259717195
     24049          77           5         -78            578   1564.397347456814   0.3694713500630989
     26041          80          11         -81            590   1640.740555354801   0.3595937200884363
     26161          77          17         -84            602   1645.260194953498   0.365899571293655
     28081          77          19         -90            622   1716.433699673115   0.3623792751904466
     28729          84          13         -85            630   1739.991914251457   0.3620706480529972
     31249          86          17         -90            674   1829.564029933067   0.3683937752234109
     33049          90          17         -91            702   1891.700622736456   0.3710946603086252
     33289          90          13         -92            714   1899.877083987447   0.3758137860694979
      discr          a           b           c         length   sqrt(d) * log(d) length/(sqrt(d)*log(d))
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  • $\begingroup$ This maybe what it looks from the data above but what I am interested is like for example if $R=13$. Then one can see $L(0,1,13)=5$ $L(5,6,13)=1$ and $L(77,16,13)=6$. Here $L(P,Q,R)$ denotes the period of continued fraction expansion of $\dfrac{P+\sqrt{R}}{Q}$. So how much we can change the value of $L(P,Q,R)$ for a fixed $R$. Whether we can increase it arbitrarily (which is not possible considering computated data above) and can we always bring it down to $1$ for some $P$ and $Q$. $\endgroup$ – Suraj Mar 10 '17 at 4:48

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