Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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!)

share|improve this question
    
Your first definition of $\xi$ is negative, but the continued fraction is for a positive irrational. –  Will Jagy Sep 3 '12 at 4:52
    
Thank you, TeX error fixed. –  Jack Huizenga Sep 3 '12 at 4:55
    
Please give your continued fractions for $D=5$ and $D=13.$ –  Will Jagy Sep 3 '12 at 4:56
2  
Oh, well. Note that if you take $\xi + 2$ you get $\frac{1 + \sqrt D}{2}.$ There really should be a standard descrition somewhere for this. in comparison, for $\sqrt D$ itself, the last $a_k = 2 a_0.$ Somebody has proved the analogous pattern for your problem. Actually, the item that gives a purely periodic continued fraction, which is what you want, is apparently $\frac{2r+1 + \sqrt D}{2}.$ See if I can prove that. –  Will Jagy Sep 3 '12 at 5:17
add comment

2 Answers 2

up vote 9 down vote accepted

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

share|improve this answer
    
Awesome, thanks! Any idea where I could find this fact? (Forgive my earlier comment, I had forgotten the term "purely periodic" includes the integer part) –  Jack Huizenga Sep 3 '12 at 5:38
1  
It should be in most accounts of continued fractions for quadratic irrationals. It's in Section 9 of the continued fractions chapter (Chapter IV) of Davenport's book The Higher Arithmetic, and it's on page 45 of the continued fractions book by Rockett and Szüsz. –  Henry Cohn Sep 3 '12 at 5:55
    
Great, thanks again! I just found it in Mollin, "Fundamental Number Theory" p. 241 as well. –  Jack Huizenga Sep 3 '12 at 5:56
add comment

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 Upper bound of period length of continued fraction representation of very composite number square root 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$ 
share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.