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Here :

http://mathworld.wolfram.com/PellEquation.html

the pell-like equation $$y^2-Dx^2=c$$ is mentioned. It is claimed that in the case $|c|<\sqrt{D}$, a solution must occur among the convergents, if I understand it right.

For the equation $$y^2-2473x^2=44$$ , this is satisfied. However, I calculated a solution of $11x^2+47xy-6y^2=44$ by using the convergents of one of the roots of $11x^2+47x-6$ and found out that

$$x=794538928954973566949076$$ $$y=15977290161340977278578$$

solves the given equation and we have $44<\sqrt{2473}$

But I did not find a convergent $\frac{p_n}{q_n}$ of $\sqrt{2473}$ satisfying $p_n^2-2473q_n^2=44$

What is wrong with the claim ? Or where is my mistake ?

  • 1
    You will find a convergent such that $\dfrac{p_n}{q_n} = \dfrac{x}{y}$. You have $p_n^2 - 2473 q_n^2 = 11$. If $c$ is not squarefree, besides coprime solutions there may be solutions with a common factor, that is the case here. – 2017-02-23
  • 0
    @DanielFischer I just noticed this, nevertheless thank you. So, if $D$ is squarefree, there is always a convergent in the case $c<\sqrt{D}$ ? – 2017-02-23
  • 0
    @DanielFischer It is mentioned that the case $c>\sqrt{D}$ is much more complicated. Do you know the method in this case ? Can we still use the convergents ? – 2017-02-23
  • 1
    For this, it's not important whether $D$ is squarefree (though generally things are nicer with squarefree $D$), but whether $c$ is. If $\lvert c\rvert < \sqrt{D}$, and $x,y$ satisfy $x^2 - Dy^2 = c$, then $\dfrac{x}{y}$ is a convergent of $\sqrt{D}$, but $x,y$ need not be coprime if $c$ is not squarefree. For the case $\lvert c\rvert > \sqrt{D}$, there are methods, but I'm not au courant with the theory of quadratic forms. – 2017-02-23
  • 0
    @DanielFischer it's not bad for Pell type. Posted answer with picture – 2017-02-23

1 Answers 1

1

Sticking with Pell type, the search for $x^2 - n y^2 = c$ is not infinite. If there is any integer solution that is primitive, meaning $\gcd(x,y) = 1,$ then the automorphism group of the form $x^2 - n y^2$ takes us to solutions that obey certain inequalities. I wrote a program based on this... note $7 \cdot 17 \cdot 23 = 2737,$ and there are eight orbits under the automorphism group.

In this case, a solution $x,y > 0,$ $x^2 - 2 y^2 = 2737$ is called a "seed" when either $$ 3x - 4 y \leq 0 \; \; \; \mbox{OR} \; \; \; -2x+3y \leq 0. $$ A solution with $x,y >0$ that is not a seed allows for a smaller positive solution (in the same orbit) $$ (3x-4y, -2x+3y) $$

If you draw some careful pictures of the lines and hyperbola involved in the inequalities, you see how this leads to bounds, which can be made explicit. As usual for Pell type, one of the inequalities (blue) is irrelevant, so there is just the blue line in the graph, no shading.

enter image description here

jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
  Automorphism matrix:  
    3   4
    2   3
  Automorphism backwards:  
    3   -4
    -2   3

  3^2 - 2 2^2 = 1

 x^2 - 2 y^2 = 2737

Thu Feb 23 12:01:21 PST 2017

x:  53  y:  6 ratio: 8.83333  SEED   KEEP +- 
x:  55  y:  12 ratio: 4.58333  SEED   KEEP +- 
x:  57  y:  16 ratio: 3.5625  SEED   KEEP +- 
x:  73  y:  36 ratio: 2.02778  SEED   KEEP +- 
x:  75  y:  38 ratio: 1.97368  SEED   BACK ONE STEP  73 ,  -36
x:  107  y:  66 ratio: 1.62121  SEED   BACK ONE STEP  57 ,  -16
x:  117  y:  74 ratio: 1.58108  SEED   BACK ONE STEP  55 ,  -12
x:  135  y:  88 ratio: 1.53409  SEED   BACK ONE STEP  53 ,  -6
x:  183  y:  124 ratio: 1.47581
x:  213  y:  146 ratio: 1.4589
x:  235  y:  162 ratio: 1.45062
x:  363  y:  254 ratio: 1.42913
x:  377  y:  264 ratio: 1.42803
x:  585  y:  412 ratio: 1.4199
x:  647  y:  456 ratio: 1.41886
x:  757  y:  534 ratio: 1.4176
x:  1045  y:  738 ratio: 1.41599
x:  1223  y:  864 ratio: 1.41551
x:  1353  y:  956 ratio: 1.41527
x:  2105  y:  1488 ratio: 1.41465
x:  2187  y:  1546 ratio: 1.41462
x:  3403  y:  2406 ratio: 1.41438
x:  3765  y:  2662 ratio: 1.41435
x:  4407  y:  3116 ratio: 1.41431
x:  6087  y:  4304 ratio: 1.41427
x:  7125  y:  5038 ratio: 1.41425
x:  7883  y:  5574 ratio: 1.41424
x:  12267  y:  8674 ratio: 1.41423
x:  12745  y:  9012 ratio: 1.41423
x:  19833  y:  14024 ratio: 1.41422
x:  21943  y:  15516 ratio: 1.41422
x:  25685  y:  18162 ratio: 1.41422
x:  35477  y:  25086 ratio: 1.41422
x:  41527  y:  29364 ratio: 1.41421
x:  45945  y:  32488 ratio: 1.41421
x:  71497  y:  50556 ratio: 1.41421
x:  74283  y:  52526 ratio: 1.41421
x:  115595  y:  81738 ratio: 1.41421
x:  127893  y:  90434 ratio: 1.41421
x:  149703  y:  105856 ratio: 1.41421
x:  206775  y:  146212 ratio: 1.41421
x:  242037  y:  171146 ratio: 1.41421
x:  267787  y:  189354 ratio: 1.41421
x:  416715  y:  294662 ratio: 1.41421
x:  432953  y:  306144 ratio: 1.41421
x:  673737  y:  476404 ratio: 1.41421
x:  745415  y:  527088 ratio: 1.41421
x:  872533  y:  616974 ratio: 1.41421
x:  1205173  y:  852186 ratio: 1.41421
x:  1410695  y:  997512 ratio: 1.41421
x:  1560777  y:  1103636 ratio: 1.41421
x:  2428793  y:  1717416 ratio: 1.41421
x:  2523435  y:  1784338 ratio: 1.41421
x:  3926827  y:  2776686 ratio: 1.41421
x:  4344597  y:  3072094 ratio: 1.41421
x:  5085495  y:  3595988 ratio: 1.41421
x:  7024263  y:  4966904 ratio: 1.41421
x:  8222133  y:  5813926 ratio: 1.41421
x:  9096875  y:  6432462 ratio: 1.41421
x:  14156043  y:  10009834 ratio: 1.41421
x:  14707657  y:  10399884 ratio: 1.41421

Thu Feb 23 12:01:41 PST 2017

 x^2 - 2 y^2 = 2737

jagy@phobeusjunior:~$

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

Notice how the other inequality becomes relevant when we switch to $x^2 - 2 y^2 = -2737.$

enter image description here

jagy@phobeusjunior:~$ ./Pell_Target_Fundamental
  Automorphism matrix:  
    3   4
    2   3
  Automorphism backwards:  
    3   -4
    -2   3

  3^2 - 2 2^2 = 1

 x^2 - 2 y^2 = -2737

Thu Feb 23 13:29:39 PST 2017

x:  1  y:  37 ratio: 0.027027  SEED   KEEP +- 
x:  25  y:  41 ratio: 0.609756  SEED   KEEP +- 
x:  31  y:  43 ratio: 0.72093  SEED   KEEP +- 
x:  41  y:  47 ratio: 0.87234  SEED   KEEP +- 
x:  65  y:  59 ratio: 1.10169  SEED   BACK ONE STEP  -41 ,  47
x:  79  y:  67 ratio: 1.1791  SEED   BACK ONE STEP  -31 ,  43
x:  89  y:  73 ratio: 1.21918  SEED   BACK ONE STEP  -25 ,  41
x:  145  y:  109 ratio: 1.33028  SEED   BACK ONE STEP  -1 ,  37
x:  151  y:  113 ratio: 1.33628
x:  239  y:  173 ratio: 1.3815
x:  265  y:  191 ratio: 1.38743
x:  311  y:  223 ratio: 1.39462
x:  431  y:  307 ratio: 1.40391
x:  505  y:  359 ratio: 1.40669
x:  559  y:  397 ratio: 1.40806
x:  871  y:  617 ratio: 1.41167
x:  905  y:  641 ratio: 1.41186
x:  1409  y:  997 ratio: 1.41324
x:  1559  y:  1103 ratio: 1.41342
x:  1825  y:  1291 ratio: 1.41363
x:  2521  y:  1783 ratio: 1.41391
x:  2951  y:  2087 ratio: 1.41399
x:  3265  y:  2309 ratio: 1.41403
x:  5081  y:  3593 ratio: 1.41414
x:  5279  y:  3733 ratio: 1.41414
x:  8215  y:  5809 ratio: 1.41418
x:  9089  y:  6427 ratio: 1.41419
x:  10639  y:  7523 ratio: 1.4142
x:  14695  y:  10391 ratio: 1.4142
x:  17201  y:  12163 ratio: 1.41421
x:  19031  y:  13457 ratio: 1.41421
x:  29615  y:  20941 ratio: 1.41421
x:  30769  y:  21757 ratio: 1.41421
x:  47881  y:  33857 ratio: 1.41421
x:  52975  y:  37459 ratio: 1.41421
x:  62009  y:  43847 ratio: 1.41421
x:  85649  y:  60563 ratio: 1.41421
x:  100255  y:  70891 ratio: 1.41421
x:  110921  y:  78433 ratio: 1.41421
x:  172609  y:  122053 ratio: 1.41421
x:  179335  y:  126809 ratio: 1.41421
x:  279071  y:  197333 ratio: 1.41421
x:  308761  y:  218327 ratio: 1.41421
x:  361415  y:  255559 ratio: 1.41421
x:  499199  y:  352987 ratio: 1.41421
x:  584329  y:  413183 ratio: 1.41421
x:  646495  y:  457141 ratio: 1.41421
x:  1006039  y:  711377 ratio: 1.41421
x:  1045241  y:  739097 ratio: 1.41421
x:  1626545  y:  1150141 ratio: 1.41421
x:  1799591  y:  1272503 ratio: 1.41421
x:  2106481  y:  1489507 ratio: 1.41421
x:  2909545  y:  2057359 ratio: 1.41421
x:  3405719  y:  2408207 ratio: 1.41421
x:  3768049  y:  2664413 ratio: 1.41421
x:  5863625  y:  4146209 ratio: 1.41421
x:  6092111  y:  4307773 ratio: 1.41421
x:  9480199  y:  6703513 ratio: 1.41421
x:  10488785  y:  7416691 ratio: 1.41421
x:  12277471  y:  8681483 ratio: 1.41421
x:  16958071  y:  11991167 ratio: 1.41421

Thu Feb 23 13:30:01 PST 2017

 x^2 - 2 y^2 = -2737

jagy@phobeusjunior:~$

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