Math Forum / Mathematics / Undergraduate Math / February 2009

JSH has given this parametric solution to x^2 - d y^2 = 1, with d = f1
f2 + 1:

x(v) = (f1 + f2 v^2)/(f1 - f2 v^2 - 2 v) - (2 d v/(f1 - f2 v^2 - 2 v) +
   1 - (f1 + f2 v^2)/(f1 - f2 v^2 - 2 v))/(d - 1)

y(v) = (2 d v/(f1 - f2 v^2 - 2 v) +
  1 - (f1 + f2 v^2)/(f1 - f2 v^2 - 2 v))/(d - 1)

(For some reason, he keeps giving those, even though they can be
considerably simplified:

  x(v) = (f1^2 - 2 f1 v + (1 + d) v^2) / (f1^2 - 2 f1 v - (-1 + d) v^2)

  y(v) = -(2 v (-f1 + v)) / (f1^2 - 2 f1 v - (-1 + d) v^2),

but I digress).

Let's look at these for d = 2, f1 = f2 = 1.  Then we have:

x(v) = (-1 + 2 v - 3 v^2) / (-1 + 2 v + v^2)
y(v) = (2 (-1 + v) v) / (-1 + 2 v + v^2)

Let's take the positive solutions of the Pell equation, x^2 - 2 y^2 = 1,
and see what values we have to set v to in order to get those solutions.

Here's the first 20 convergents of the continued fraction for sqrt(2):

1
3/2
7/5
17/12
41/29
99/70
239/169
577/408
1393/985
3363/2378
8119/5741
19601/13860
47321/33461
114243/80782
275807/195025
665857/470832
1607521/1136689
3880899/2744210
9369319/6625109
22619537/15994428

Every other one of these, starting at the second, corresponds to a
solution to the Pell equation.  Here are those solutions, in the form

  x,y {{v->n/m}}

which means that the solution x, y is found by setting v to n/m in JSH's
parametric solution.

3,2 {{v->1/3}}
17,12 {{v->2/5}}
99,70 {{v->7/17}}
577,408 {{v->12/29}}
3363,2378 {{v->41/99}}
19601,13860 {{v->70/169}}
114243,80782 {{v->239/577}}
665857,470832 {{v->408/985}}
3880899,2744210 {{v->1393/3363}}
22619537,15994428 {{v->2378/5741}}

Notice anything interesting there?  Take a look at the numerators and
denominators of the v's, and compare to the convergents (all the
convergents, not just the half of them that give solutions).

If Pi/Qi is the i'th convergent, the v's are

  P1/P2
  Q2/Q3
  P3/P4
  Q4/Q5
  ...

I think this is a very pretty pattern.  It's neat the way the v's are
formed from ratios of the numerators of successive convergents
alternating with the ratios of denominators of successive convergents.

I don't see any obvious pattern like that for any other d I've tried (3,
5, 15, using every possible f1 for each), so this appears like it might
be a property just of 2 (or sqrt(2), I suppose, depending on how you
look at it).

Here's the Mathematica code I used to play around with this:

ClearAll["Global`*"];
y[v_] := (2 d v/(f1 - f2 v^2 - 2 v) +
   1 - (f1 + f2 v^2)/(f1 - f2 v^2 - 2 v))/(d - 1)
x[v_] := (f1 + f2 v^2)/(f1 - f2 v^2 -
    2 v) - (2 d v/(f1 - f2 v^2 - 2 v) +
    1 - (f1 + f2 v^2)/(f1 - f2 v^2 - 2 v))/(d - 1)
d = 2;
f1 = 1;
f2 = (d - 1)/f1;
cv = Convergents[Sqrt[d], 20]
Numerator[cv]^2 - d Denominator[cv]^2
For[i = 2, i <= 20, i += 2, tx = Numerator[cv[[i]]];
ty = Denominator[cv[[i]]];
Print[tx, ",", ty, Solve[{x[v] == tx, y[v] == ty}, v]]]