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I worked out the periodicity of some infinite continued fractions last night by hand. (Don't ask me why)

For example, $\sqrt{13}= [3,1,1,1,1,6,1,1,1,1,6,\ldots]$. Last night I worked out the first period of this continued fraction and the algebra was a little meh. I was wondering, what is the largest continued fraction period ever worked out by hand before?

For example: $\sqrt{D}$ may have the continued fraction expansion: $[\text{repeat}(a_1,a_2,a_3,\ldots, a_n)]$. Define the "first period worked out by hand" to be:

The discovery of the first $a_1,a_2,a_3,\ldots,a_n$ of the infinite continued fraction $\sqrt{D}$ using nothing but pencil, and paper.

Any stories for me?

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  • $\begingroup$ No story, but there are efficient ways to compute the continued fraction of square roots. $\endgroup$ – André Nicolas Jul 19 '13 at 23:08
  • $\begingroup$ How is that related to math history? $\endgroup$ – lhf Jul 20 '13 at 1:35
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Lagrange's method uses just integer arithmetic and is suitable for use by hand. See How to detect when continued fractions period terminates

If you need more detail let me know.

Note that I used precisely that in Minimum of $n$? $123456789x^2 - 987654321y^2 =n$ ($x$,$y$ and $n$ are positive integers) although it was by computer.

Not by the way, if you are primarily interested in the square root of positive a integer $D,$ then the triple indicating a first form in the cycle is given by finding $$ a_0 = \lfloor \sqrt D \rfloor $$ and then forming the triple $$ \langle 1, 2 a_0, a_0^2 - D \rangle $$

Here, the triple $ \langle a, b, c \rangle $ refers to the quadratic form $$ f(x,y) = a x^2 + b x y + c y^2. $$ The form is "reduced" if both $ac <0$ and $b > |a+c|.$

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./Pell
Input n for Pell 
991

0  form   1 62 -30   delta  -2
1  form   -30 58 5   delta  12
2  form   5 62 -6   delta  -10
3  form   -6 58 25   delta  2
4  form   25 42 -22   delta  -2
5  form   -22 46 21   delta  2
6  form   21 38 -30   delta  -1
7  form   -30 22 29   delta  1
8  form   29 36 -23   delta  -2
9  form   -23 56 9   delta  6
10  form   9 52 -35   delta  -1
11  form   -35 18 26   delta  1
12  form   26 34 -27   delta  -1
13  form   -27 20 33   delta  1
14  form   33 46 -14   delta  -3
15  form   -14 38 45   delta  1
16  form   45 52 -7   delta  -8
17  form   -7 60 13   delta  4
18  form   13 44 -39   delta  -1
19  form   -39 34 18   delta  2
20  form   18 38 -35   delta  -1
21  form   -35 32 21   delta  2
22  form   21 52 -15   delta  -3
23  form   -15 38 42   delta  1
24  form   42 46 -11   delta  -4
25  form   -11 42 50   delta  1
26  form   50 58 -3   delta  -20
27  form   -3 62 10   delta  6
28  form   10 58 -15   delta  -4
29  form   -15 62 2   delta  31
30  form   2 62 -15   delta  -4
31  form   -15 58 10   delta  6
32  form   10 62 -3   delta  -20
33  form   -3 58 50   delta  1
34  form   50 42 -11   delta  -4
35  form   -11 46 42   delta  1
36  form   42 38 -15   delta  -3
37  form   -15 52 21   delta  2
38  form   21 32 -35   delta  -1
39  form   -35 38 18   delta  2
40  form   18 34 -39   delta  -1
41  form   -39 44 13   delta  4
42  form   13 60 -7   delta  -8
43  form   -7 52 45   delta  1
44  form   45 38 -14   delta  -3
45  form   -14 46 33   delta  1
46  form   33 20 -27   delta  -1
47  form   -27 34 26   delta  1
48  form   26 18 -35   delta  -1
49  form   -35 52 9   delta  6
50  form   9 56 -23   delta  -2
51  form   -23 36 29   delta  1
52  form   29 22 -30   delta  -1
53  form   -30 38 21   delta  2
54  form   21 46 -22   delta  -2
55  form   -22 42 25   delta  2
56  form   25 58 -6   delta  -10
57  form   -6 62 5   delta  12
58  form   5 58 -30   delta  -2
59  form   -30 62 1   delta  62
60  form   1 62 -30

 disc   3964
Automorph, written on right of Gram matrix:  
5788591406539787767296194303  361672073709940783423276163010
12055735790331359447442538767  753244210407084073508733597857


 Pell automorph 
379516400906811930638014896080  11947234168218377212415555918097
12055735790331359447442538767  379516400906811930638014896080

Pell unit 
379516400906811930638014896080^2 - 991 * 12055735790331359447442538767^2 = 1 

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

991       991

jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ date
Fri Jul 19 16:34:12 PDT 2013
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ 
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