Pell's Equation
== Definition == The Diophantine equation: :$x^2 - n y^2 = 1$ is known as '''Pell's equation'''. == Solution == 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 quotients. The last element $x$ is of course not an integer. What we do have, though, is: {{begin-eqn}} {{eqn | l = x       | r = \sqbrk {\sequence {2 a, b_1, b_2, \ldots, b_2, b_1} }       | c =  }} {{eqn | r = a + \sqbrk {a, \sequence {b_1, b_2, \ldots, b_2, b_1, 2 a} }       | c =  }} {{eqn | r = a + \sqrt n       | c =  }} {{end-eqn}} 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  of this is an integer while the  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: {{begin-eqn}} {{eqn | n = 1       | l = a q_{r s} + q_{r s - 1}       | r = p_{r s} }} {{eqn | n = 2       | l = a p_{r s} + p_{r s - 1}       | r = n q_{r s} }} {{end-eqn}} 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  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.  == Sequence == {{:Pell's Equation/Sequence}} == Examples == {{:Pell's Equation/Examples}} == Also known as == A particular instance of '''Pell's equation''' can also be seen referred to as '''a Pellian equation'''. {{Namedfor|John Pell|cat = Pell}} == Historical Note == {{:Pell's Equation/Historical Note}} == Sources == * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = Best Rational Approximations to Root 2 generate Pythagorean Triples|next = Sprague's Property of Root 2}}: $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$ * {{BookReference|Curious and Interesting Numbers|1986|David Wells|prev = 4,729,494|next = Pell's Equation/Historical Note}}: $4,729,494$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Best Rational Approximations to Root 2 generate Pythagorean Triples|next = Square Root of 2 as Sum of Egyptian Fractions}}: $1 \cdotp 41421 \, 35623 \, 73095 \, 04880 \, 16887 \, 24209 \, 69807 \, 85697 \ldots$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = Definition:Emirp/Sequence|next = Pell's Equation/Examples/13}}: $13$ * {{BookReference|Curious and Interesting Numbers|1997|David Wells|ed = 2nd|edpage = Second Edition|prev = 4,729,494|next = Pell's Equation/Historical Note}}: $4,729,494$ * {{BookReference|The Penguin Dictionary of Mathematics|1998|David Nelson|ed = 2nd|edpage = Second Edition|prev = Mathematician:Charles Sanders Peirce|next = Definition:Pencil|entry = Pell's equation}} * {{BookReference|The Penguin Dictionary of Mathematics|2008|David Nelson|ed = 4th|edpage = Fourth Edition|prev = Mathematician:Charles Sanders Peirce|next = Definition:Pencil|entry = Pell's equation}} * {{BookReference|Taming the Infinite|2008|Ian Stewart|prev = Mathematician:Pierre de Fermat|next = Mathematician:Carl Friedrich Gauss}}: Chapter $7$: Patterns in Numbers: Fermat * {{BookReference|The Concise Oxford Dictionary of Mathematics|2014|Christopher Clapham|author2 = James Nicholson|ed = 5th|edpage = Fifth Edition|prev = Mathematician:John Pell|next = Definition:Pendulum|entry = Pell's equation}} * {{MathWorld|Pell Equation|PellEquation}}