Solving Diophantine Equation - odd Periods

Question

I am trying to solve the Diophantine equation using continuous fraction .

x ^ 2 - D * Y ^ 2 = 1

Keeping this document as reference

http://library.msri.org/books/Book44/files/01lenstra.pdf

In the page No. 3 , we have an example for Number 14.

If we write continuous fraction for 14 we get

[3; 1, 2, 1, 6]

Now , writing it as 3 + 1 / 1 + 1/ 2 + 1/ 1 + 1 / 6

once i derive i get , 15/ 4 which solves pells equation as

15 ^ 2 = 14 * 4 ^ 2 + 1

This is for even period and it works well.

Last line in the doc is given as

"If the period length is even, one proceeds as above; if the period length is odd, one truncates at the end of the second period [Buhler and Wagon 2008]."

But when i try for odd period , i dont get the correct answer ..

i took , 29 as the number

[5; 2, 1, 1, 2, 10]

when written as fraction , i get 727/135 which doesnt solve the equation

and i tried [5; 2, 1, 1, 2] , i got 70/29 which doesnt solve

and then tried [5; 2, 1, 1] i got 27/5. Even that doesnt solve

am i missing something ???

Answer 1

This is how I like to write these out. The value of $x^2 - d y^2$ for $x=p$ and $y=q$ is written directly below each convergent $\frac{p}{q}.$ I recommend that you practice writing yours the same way, it helps to see that the values of $x^2 - d y^2$ also repeat in absolute value but have a $\pm$ switch when there does exist a solution to $x^2 - d y^2 = -1;$ so first we find $70^2 - 29 \cdot 13^2 = -1,$ only later do we find $9801^2 - 29 \cdot 1820^2 = 1.$

$$ \sqrt {29} $$

$$ \begin{array}{cccccccccccccccccccccccccccccc} & & 5 & & 2 & & 1 & & 1 & & 2 & & 10 & & 2 & & 1 & & 1 & & 2 & & 10 & \\ \frac{0}{1} & \frac{1}{0} & & \frac{5}{1} & & \frac{11}{2} & & \frac{16}{3} & & \frac{27}{5} & & \frac{70}{13} & & \frac{727}{135} & & \frac{1524}{283} & & \frac{2251}{418} & & \frac{3775}{701} & & \frac{9801}{1820} & & \frac{101785}{18901} \\ \\ -29 & 1 & & -4 & & 5 & & -5 & & 4 & & -1 & & 4 & & -5 & & 5 & & -4 & & 1 & & -4 \end{array} $$ We begin with two fake "convergents," first "$p/q = 0/1,$" and below that $0^2 - 29 \cdot 1^2 = -29.$ Next, "$p/q = 1/0,$" and below that $1^2 - 29 \cdot 0^2 = 1.$

Then our first genuine convergent, $p/q = 5/1,$ and below that $5^2 - 29 \cdot 1^2 = -4.$ Next, $p/q = 11/2,$ and below that $11^2 - 29 \cdot 2^2 = 5.$

A few steps later, $p/q = 70/13,$ and below that $70^2 - 29 \cdot 13^2 = -1.$

Several more steps later, $p/q = 9801/1820,$ and below that $9801^2 - 29 \cdot 1820^2 = 1.$

Smaller displays so that the entire calculation might fit in the intended width:

First we use the command "small:"

$$ \small \begin{array}{cccccccccccccccccccccccccccccc} & & 5 & & 2 & & 1 & & 1 & & 2 & & 10 & & 2 & & 1 & & 1 & & 2 & & 10 & \\ \frac{0}{1} & \frac{1}{0} & & \frac{5}{1} & & \frac{11}{2} & & \frac{16}{3} & & \frac{27}{5} & & \frac{70}{13} & & \frac{727}{135} & & \frac{1524}{283} & & \frac{2251}{418} & & \frac{3775}{701} & & \frac{9801}{1820} & & \frac{101785}{18901} \\ \\ -29 & 1 & & -4 & & 5 & & -5 & & 4 & & -1 & & 4 & & -5 & & 5 & & -4 & & 1 & & -4 \end{array} $$

now scriptsize

$$ \scriptsize \begin{array}{cccccccccccccccccccccccccccccc} & & 5 & & 2 & & 1 & & 1 & & 2 & & 10 & & 2 & & 1 & & 1 & & 2 & & 10 & \\ \frac{0}{1} & \frac{1}{0} & & \frac{5}{1} & & \frac{11}{2} & & \frac{16}{3} & & \frac{27}{5} & & \frac{70}{13} & & \frac{727}{135} & & \frac{1524}{283} & & \frac{2251}{418} & & \frac{3775}{701} & & \frac{9801}{1820} & & \frac{101785}{18901} \\ \\ -29 & 1 & & -4 & & 5 & & -5 & & 4 & & -1 & & 4 & & -5 & & 5 & & -4 & & 1 & & -4 \end{array} $$

Next we use "tiny"

$$ \tiny \begin{array}{cccccccccccccccccccccccccccccc} & & 5 & & 2 & & 1 & & 1 & & 2 & & 10 & & 2 & & 1 & & 1 & & 2 & & 10 & \\ \frac{0}{1} & \frac{1}{0} & & \frac{5}{1} & & \frac{11}{2} & & \frac{16}{3} & & \frac{27}{5} & & \frac{70}{13} & & \frac{727}{135} & & \frac{1524}{283} & & \frac{2251}{418} & & \frac{3775}{701} & & \frac{9801}{1820} & & \frac{101785}{18901} \\ \\ -29 & 1 & & -4 & & 5 & & -5 & & 4 & & -1 & & 4 & & -5 & & 5 & & -4 & & 1 & & -4 \end{array} $$

That is pretty good, please concentrate on the line of $x^2 - d y^2$ values, maybe ignore the initial $-29,$ so $$ 1, \; -4, \; 5, \; -5, \; 4, \; -1, \; 4, \; -5, \; 5, \; -4, \; 1, \; -4, \ldots $$

I got instructions on re-sizing Latex at What are some options for putting another, smaller copy of this long Latex item?