Solution to Pell's Equation

Theorem

Recall Pell's equation:

The Diophantine equation:

$x^2 - n y^2 = \pm 1$

is known as Pell's equation.


Let the continued fraction of $\sqrt n$ have a cycle whose length is $s$:

$\sqrt n = \sqbrk {a_1 \sequence {a_2, a_3, \ldots, a_{s + 1} } }$

Let $a_n = \dfrac {p_n} {q_n}$ be a convergent of $\sqrt n$.

Then:

${p_{r s} }^2 - n {q_{r s} }^2 = \paren {-1}^{r s}$ for $r = 1, 2, 3, \ldots$

and all solutions of:

$x^2 - n y^2 = \pm 1$

are given in this way.


Proof

First note that if $x = p, y = q$ is a positive solution of $x^2 - n y^2 = 1$ then $\dfrac p q$ is a convergent of $\sqrt n$.


The continued fraction of $\sqrt n$ is periodic from Continued Fraction Expansion of Irrational Square Root and of the form:

$\sqbrk {a \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a} }$

or

$\sqbrk {a \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a} }$

For each $r \ge 1$ we can write $\sqrt n$ as the (non-simple) finite continued fraction:

$\sqrt n = \sqbrk {a \sequence {b_1, b_2, \ldots, b_2, b_1, 2 a, b_1, b_2, \ldots, b_2, b_1, x} }$

which has a total of $r s + 1$ partial denominators.

The last element $x$ is of course not an integer.


What we do have, though, is:

\(\ds x\) \(=\) \(\ds \sqbrk {\sequence {2 a, b_1, b_2, \ldots, b_2, b_1} }\)
\(\ds \) \(=\) \(\ds a + \sqbrk {a, \sequence {b_1, b_2, \ldots, b_2, b_1, 2 a} }\)
\(\ds \) \(=\) \(\ds a + \sqrt n\)


The final three convergents in the above FCF are:

$\dfrac {p_{r s - 1} } {q_{r s - 1} }, \quad \dfrac {p_{r s} } {q_{r s} }, \quad \dfrac {x p_{r s} + p_{r s - 1} } {x q_{r s} + q_{r s - 1} }$

The last one of these equals $\sqrt n$ itself.

So:

$\sqrt n \paren {x q_{r s} + q_{r s - 1} } = \paren {x p_{r s} + p_{r s - 1} }$

Substituting $a + \sqrt n$ for $x$, we get:

$\sqrt n \paren {\paren {a + \sqrt n} q_{r s} + q_{r s - 1} } = \paren {\paren {a + \sqrt n} p_{r s} + p_{r s - 1} }$

This simplifies to:

$\sqrt n \paren {a q_{r s} + q_{r s - 1} - p_{r s} } = a p_{r s} + p_{r s - 1} - n q_{r s}$

The right hand side of this is an integer while the left hand side is $\sqrt n$ times an integer.

Since $\sqrt n$ is irrational, the only way that can happen is if both sides equal zero.

This gives us:

\(\text {(1)}: \quad\) \(\ds a q_{r s} + q_{r s - 1}\) \(=\) \(\ds p_{r s}\)
\(\text {(2)}: \quad\) \(\ds a p_{r s} + p_{r s - 1}\) \(=\) \(\ds n q_{r s}\)

Multiplying $(1)$ by $p_{r s}$, $(2)$ by $q_{r s}$ and then subtracting:

$p_{r s}^2 - n q_{r s}^2 = p_{r s} q_{r s - 1} - p_{r s - 1} q_{r s}$

By Difference between Adjacent Convergents of Simple Continued Fraction, the right hand side of this is $\paren {-1}^{r s}$.


When the cycle length $s$ of the continued fraction of $\sqrt n$ is even, we have $\paren {-1}^{r s} = 1$.

Hence $x = p_{r s}, y = q_{r s}$ is a solution to Pell's equation for each $r \ge 1$.

When $s$ is odd, though:

$x = p_{r s}, y = q_{r s}$ is a solution of $x^2 - n y^2 = -1$ when $r$ is odd
$x = p_{r s}, y = q_{r s}$ is a solution of $x^2 - n y^2 = 1$ when $r$ is even.

$\blacksquare$


Sequence

The minimum solutions of $x$ and $y$ for the Diophantine equation $x^2 - n y^2 = 1$ for various values of $n$ are as follows:

$n$ $x$ $y$
$2$ $3$ $2$
$3$ $2$ $1$
$5$ $9$ $4$
$6$ $5$ $2$
$7$ $8$ $3$
$8$ $3$ $1$
$10$ $19$ $6$
$11$ $10$ $3$
$12$ $7$ $2$
$13$ $649$ $180$
$14$ $15$ $4$
$15$ $4$ $1$

Note the unusual peak at $13$.


Thus, the sequence for $x$ for non-square $n$ begins:

$3, 2, 9, 5, 8, 3, 19, 10, 7, 649, 15, 4, 33, 17, 170, \ldots$


Similarly, the sequence for $y$ for non-square $n$ begins:

$2, 1, 4, 2, 3, 1, 6, 3, 2, 180, 4, 1, 8, 4, 39, 2, 12, \ldots$


Examples

Pell's equation: $x^2 - 2 y^2 = 1$

$x^2 - 2 y^2 = 1$

has the positive integral solutions:

$\begin {array} {r|r} x & y \\ \hline 3 & 2 \\ 17 & 12 \\ 99 & 70 \\ 577 & 408 \\ 3363 & 2378 \\ \end {array}$

and so on.


Pell's equation: $x^2 - 2 y^2 = -1$

$x^2 - 2 y^2 = -1$

has the positive integral solutions:

$\begin {array} {r|r} x & y \\ \hline 1 & 1 \\ 7 & 5 \\ 41 & 29 \\ 239 & 169 \\ 1393 & 985 \\ \end {array}$

and so on.


Pell's equation: $x^2 - 8 y^2 = 1$

$x^2 - 8 y^2 = 1$

has the positive integral solutions:

\(\ds \tuple {x, y}\) \(=\) \(\ds \tuple {3, 1}\)
\(\ds \tuple {x, y}\) \(=\) \(\ds \tuple {17, 6}\)
\(\ds \tuple {x, y}\) \(=\) \(\ds \tuple {99, 35}\)
\(\ds \tuple {x, y}\) \(=\) \(\ds \tuple {577, 204}\)
\(\ds \tuple {x, y}\) \(=\) \(\ds \tuple {3363, 1189}\)

and so on.


Pell's equation: $x^2 - 13 y^2 = 1$

$x^2 - 13 y^2 = 1$

has the smallest positive integral solution:

$x = 649$
$y = 180$


Pell's equation: $x^2 - 29 y^2 = 1$

$x^2 - 29 y^2 = 1$

has the smallest positive integral solution:

$x = 9801$
$y = 1820$


Pell's equation: $x^2 - 61 y^2 = 1$

$x^2 - 61 y^2 = 1$

has the smallest positive integral solution:

$x = 1 \, 766 \, 319 \, 049$
$y = 226 \, 153 \, 980$


Source of Name

This entry was named for John Pell.


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Pell's equation
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Pell's equation
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