Question

Once again, working with stable vector bundles on $\mathbb{P}^2$ I have run into a question that is really out of my area. (Thanks to everybody who helped out with my last question!)

Let $D>9$ be a rational number which is not a square, and consider the quadratic irrational

$$\xi = \frac{-3 + \sqrt{D}}{2},$$

(I'd be willing to force $D$ to be an integer, and even to assume $D \equiv 5 \pmod{8}$, but I don't think it matters). Let $r$ be the positive integer such that

$$(2r+1)^2 < D < (2r+3)^2.$$

Numerous examples with Mathematica suggest that the continued fraction expansion of $\xi$ takes the form

$$\xi = [r-1;\overline{a_1,\ldots,a_k}],$$ where the last term of the repeating part is $a_k = 2r+1$.

For my particular situtation, I'd be happy enough to know that the number $2r+1$ appears somewhere in the expansion.

As I know almost nothing about continued fractions aside from statements of the basic results, I haven't the slightest idea how to prove something like this. I also don't have a source which does much more advanced things than show that the expansion of a quadratic irrational always repeats. Are statements like this well-known? And where should I look for more advanced theory relevant to this problem?

In case it helps, the original form I came to this number is as follows. Put $$ q = \frac{1}{8}(D-5). $$ Then $\xi$ is a solution of the equation $$\frac{1}{2}(x^2+3x+1) = q.$$ Thanks!

EDIT: At request, here is what Mathematica gives for the continued fractions for some $D$:

$D=5: [0;-2,\overline{-1}]$ (but I am requiring $D>9$)

$D=10: [0;\overline{12,3}]$

$D=13: [0;\overline{3}]$

$D=141: [4;\overline{2,3,2,11}]$

(need more examples? Just ask!)

Answer 1

It's known that a quadratic irrational has a purely periodic continued fraction expansion if and only if it is greater than $1$ and its conjugate is between $-1$ and $0$. Your observation amounts to proving that $r+2 + (-3+\sqrt{D})/2$ has this property (note that adding $r+2$ makes it start with $2r+1$). That amounts to checking that $-1 < (2r+1 - \sqrt{D})/2 < 0$, which is equivalent to your assumption that $(2r+1)^2 < D < (2r+3)^3$.

Answer 2

EDIT: I got your fact right here.

ORIGINAL: It seems Henry got it. Meanwhile, let me point out how things appear from the Lagrange viewpoint of right-adjacent reduced forms: given odd numbers $n$ and $1 \leq m \leq n,$ the cycle for the form $\langle -1, n, m \rangle $ has penultimate form $\langle m, n, -1 \rangle, $ then "digit" $\delta = -n,$ then the end of the cycle is again $\langle -1, n, m \rangle .$ Well, see the method in my answer at http://mathoverflow.net/questions/22811/upper-bound-of-period-length-of-continued-fraction-representation-of-very-composi where the fact you need, the final $\delta = -n,$ follows from the definition of the $\delta$'s.

Examples:

=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 5 1

  0  form             -1           5           1


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 5 1   delta  5
1  form   1 5 -1   delta  -5
2  form   -1 5 1
minimum was   1rep 1 0 disc   29 dSqrt 5.3851648071  M_Ratio  29
Automorph, written on right of Gram matrix:  
-1  5
5  -26
 Trace:  -27   gcd(a21, a22 - a11, a12) : 5
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 5 3

  0  form             -1           5           3


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 5 3   delta  1
1  form   3 1 -3   delta  -1
2  form   -3 5 1   delta  5
3  form   1 5 -3   delta  -1
4  form   -3 1 3   delta  1
5  form   3 5 -1   delta  -5
6  form   -1 5 3
minimum was   1rep 1 0 disc   37 dSqrt 6.0827625303  M_Ratio  37
Automorph, written on right of Gram matrix:  
-13  72
24  -133
 Trace:  -146   gcd(a21, a22 - a11, a12) : 24
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 5 5

  0  form             -1           5           5


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 5 5   delta  1
1  form   5 5 -1   delta  -5
2  form   -1 5 5
minimum was   1rep 1 0 disc   45 dSqrt 6.7082039325  M_Ratio  45
Automorph, written on right of Gram matrix:  
-1  5
1  -6
 Trace:  -7   gcd(a21, a22 - a11, a12) : 1
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 7 1

  0  form             -1           7           1


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 7 1   delta  7
1  form   1 7 -1   delta  -7
2  form   -1 7 1
minimum was   1rep 1 0 disc   53 dSqrt 7.2801098893  M_Ratio  53
Automorph, written on right of Gram matrix:  
-1  7
7  -50
 Trace:  -51   gcd(a21, a22 - a11, a12) : 7
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 7 3

  0  form             -1           7           3


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 7 3   delta  2
1  form   3 5 -3   delta  -2
2  form   -3 7 1   delta  7
3  form   1 7 -3   delta  -2
4  form   -3 5 3   delta  2
5  form   3 7 -1   delta  -7
6  form   -1 7 3
minimum was   1rep 1 0 disc   61 dSqrt 7.8102496759  M_Ratio  61
Automorph, written on right of Gram matrix:  
-79  585
195  -1444
 Trace:  -1523   gcd(a21, a22 - a11, a12) : 195
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 7 5

  0  form             -1           7           5


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 7 5   delta  1
1  form   5 3 -3   delta  -1
2  form   -3 3 5   delta  1
3  form   5 7 -1   delta  -7
4  form   -1 7 5
minimum was   1rep 1 0 disc   69 dSqrt 8.3066238629  M_Ratio  69
Automorph, written on right of Gram matrix:  
2  -15
-3  23
 Trace:  25   gcd(a21, a22 - a11, a12) : 3
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$ ./indefCycle
Input three coefficients a b c for indef f(x,y)= a x^2 + b x y + c y^2 
-1 7 7

  0  form             -1           7           7


           1           0
           0           1

To Return  
           1           0
           0           1

0  form   -1 7 7   delta  1
1  form   7 7 -1   delta  -7
2  form   -1 7 7
minimum was   1rep 1 0 disc   77 dSqrt 8.7749643874  M_Ratio  77
Automorph, written on right of Gram matrix:  
-1  7
1  -8
 Trace:  -9   gcd(a21, a22 - a11, a12) : 1
=========================================
jagy@phobeusjunior:~/old drive/home/jagy/Cplusplus$