Pell Equation

Summary: The Pell equation $m^2=a{n}^2+1$ (or equivalently $m^2-a{n}^2=1$) is a Diophantine equation whose integer solutions are the key ingredient in generating infinite families of integer solutions to $a{x}^2+b=y^2$; Euler describes Pell’s iterative descent method for finding the minimal positive solution.

Sources: chapter-2.0.6, chapter-2.0.7, chapter-2.0.11, chapter-2.0.12, chapter-2.0.13, additions-2, additions-7, additions-8

Last updated: 2026-05-10

---

Definition

For a given integer $a$ that is positive and not a perfect square, the Pell equation is:

$m^2=a{n}^2+1(m^2-a{n}^2=1)$

The trivial solution $m=1$, $n=0$ always exists. The problem is to find the minimal positive solution $(m,n)$ with $n>0$.

---

Why It Matters

The Pell equation is the engine of ch2.0.6-integer-solutions-quadratic-squares: if $(f,g)$ is any integer solution to $a{f}^2+b=g^2$, and $(m,n)$ is any solution to $m^2=a{n}^2+1$, then

$x=mf+ng,y=mg+anf$

is also an integer solution to $a{x}^2+b=y^2$. Iterating this map from a single seed $(f,g)$ produces an infinite sequence of solutions. (source: chapter-2.0.6, §82–86)

---

Exclusion Conditions (§97)

The equation $m^2=a{n}^2+1$ has no positive integer solution $(n>0)$ when:

• $a<0$: The right side would grow too fast for fixed $m$.
• $a$ is a perfect square: $a{n}^2$ is then a perfect square, and no perfect square can be increased by 1 to yield another perfect square in integers. (If $a{n}^2=k^2$ then $k^2+1$ is between two consecutive squares $k^2$ and $(k+1{)}^2$ for $k\ge 1$.)

For all other positive integers $a$, a solution always exists.

---

Pell’s Method (§98–104)

Attribute: Euler credits this to John Pell (1611–1685), an English mathematician. The name has stuck, although the equation predates Pell.

Algorithm (for a given positive non-square $a$):

1. Let $e=\lfloor \sqrt{a}\rfloor$. The root $m$ satisfies $en<m<(e+1)n$ for $n>0$.
2. Write $m=en+p$ or $m=en-p$ depending on whether $m>en$ or $m<en$.
3. Square, simplify; express $n$ in terms of $p$ and $\sqrt{(\text{something})p^2+(\text{something})}$.
4. Bound $n$ relative to $p$ to pick the next substitution $n=kp+q$ (for appropriate integer $k$).
5. Repeat until a radical $\sqrt{a{t}^2+1}$ appears — the same structure as the original problem.
6. Set $t=0$, recover the base values, and trace back through the chain.

The method always terminates because the sequence of “active variables” is strictly decreasing at each step (§104).

---

Closed-Form Solutions Near Perfect Squares (§107–111)

$a$	$n$	$m$	Identity used
$e^2-2$	$e$	$e^2-1$	$(e^2-2)e^2+1=(e^2-1{)}^2$
$e^2-1$	$2e$	$2e^2-1$	$(e^2-1)(4e^2)+1=(2e^2-1{)}^2$
$e^2+1$	$2e$	$2e^2+1$	$(e^2+1)(4e^2)+1=(2e^2+1{)}^2$
$e^2+2$	$e$	$e^2+1$	$(e^2+2)e^2+1=(e^2+1{)}^2$

---

Recurrence from the Minimal Solution (§95)

Once the minimal Pell pair $(m,n)$ is known, the sequence of $x$-values satisfying $a{x}^2+b=y^2$ obeys the linear recurrence

$x_{k+2}=2m\cdot x_{k+1}-x_k,$

and the same recurrence holds for the $y$-values independently. This means only the first two terms (the seed $f$ and the first new value $A=ng+mf$) are needed to extend the sequence indefinitely.

---

Selected Minimal Solutions

$a$	$n$	$m$
2	2	3
3	1	2
5	4	9
7	3	8
13	180	649
29	1820	9801
61	226153980	1766319049

The extreme case $a=61$ is famous: its minimal solution has $n\approx 2.26\times 10^8$.

---

---

Lagrange’s Existence Proof (Add. II, Art. 37)

Euler’s iterative method finds the minimal Pell solution but assumes one exists. The first rigorous proof of existence is Lagrange’s, in Additions Chapter II:

Theorem: For every positive non-square integer $K$, the equation $p^2-K{q}^2=1$ has a positive integer solution.

Proof: Apply the binary-quadratic-form algorithm of Add. II, Art. 33 (see binary-quadratic-forms) to $F(x,1)=x^2-K$, generating the auxiliary sequences $P^0=1,Q^0=0$, $\mu <\sqrt{K},\dots$ The fundamental identity $Q^{k2}-P^{k-1}{P}^k=K$ together with the alternating-sign argument (Add. II, Art. 34) bounds $|P^k|\le 4K$ and $|Q^k|\le 2\sqrt{K}$. Pigeonhole on the finite integer lattice forces $(P^{\pi },Q^{\pi })$ to recur; the period $2\rho$ closes with $P^{2\rho }=P^0=1$, giving

$p^{2\rho 2}-K{q}^{2\rho 2}=1.$

This is the desired Pell solution. ∎

See periodicity-quadratic-irrationals for the underlying periodicity theorem.

---

Convergent Characterization (Add. II, Art. 38)

Theorem: If $p^2-K{q}^2=\pm H$ with $H<\sqrt{K}$ has a positive integer solution, then $p/q$ must be a principal convergent of $\sqrt{K}$.

Proof sketch: From $p^2-K{q}^2=H$, factor as $(p-q\sqrt{K})(p+q\sqrt{K})=H$. Then

$\left|p-q\sqrt{K}\right|=\frac{H}{p+q\sqrt{K}}<\frac{1}{2q}$

(using $p/q>\sqrt{K}$ and $H<\sqrt{K}$). This is the Hurwitz-style criterion: any rational $p/q$ with $|p/q-\sqrt{K}|<1/(2q^2)$ is necessarily a principal convergent of $\sqrt{K}$. ∎

Consequence: To search for solutions of $p^2-K{q}^2=\pm H$ with $H$ small, one only needs to scan the CF expansion of $\sqrt{K}$ and check each convergent — a far more efficient search than brute force. The full table of CFs for $\sqrt{N}$ ($N\le 99$) is in square-root-continued-fractions.

---

Composition Law (Brahmagupta’s Bhāvanā)

The set of solutions $(m,n)$ to $m^2-a{n}^2=1$ is multiplicatively closed under the composition law from the brahmagupta-fibonacci-identity (with $c=-a$):

$(m_1^2-a{n}_1^2)(m_2^2-a{n}_2^2)=(m_1{m}_2+a{n}_1{n}_2{)}^2-a(m_1{n}_2+m_2{n}_1{)}^2.$

So if $(m_1,n_1)$ and $(m_2,n_2)$ both satisfy the Pell equation, then $(m_1{m}_2+a{n}_1{n}_2,m_1{n}_2+m_2{n}_1)$ does too. Iterating from the minimal solution generates all solutions — the standard modern approach. Euler hints at this in ch2.0.11-quadratic-form-factorization §176.

This composition also explains the surprise of ch2.0.13-impossibility-biquadrate-sums §211: $x^4-2y^4=\square$ has infinitely many solutions because $r^2-10s^2=1$ (a Pell-type auxiliary) has infinitely many.

---

Closed Form via Powers (Add. VII, Art. 75)

Let $(T,V)$ be the least positive solution of $t^2-A{u}^2=1$. Then all integer solutions are $t\pm u\sqrt{A}=(T\pm V\sqrt{A}{)}^m,m=1,2,3,\dots$ or, separating rational and irrational parts, $t=\frac{(T+V\sqrt{A}{)}^m+(T-V\sqrt{A}{)}^m}{2},u=\frac{(T+V\sqrt{A}{)}^m-(T-V\sqrt{A}{)}^m}{2\sqrt{A}}.$ Lagrange proves this exhausts all solutions: any putative intermediate $(\theta ,v)$ between consecutive powers $m$ and $m+1$ would yield a $u<V$, contradicting minimality.

The binomial expansion gives $t=T^m+\left(\genfrac{}{}{0ex}{}{m}{2}\right)A{T}^{m-2}{V}^2+\left(\genfrac{}{}{0ex}{}{m}{4}\right)A^2{T}^{m-4}{V}^4+\cdots$ $u=m{T}^{m-1}V+\left(\genfrac{}{}{0ex}{}{m}{3}\right)A{T}^{m-3}{V}^3+\cdots$ which is the same as the ch2.0.12-quadratic-form-as-power machinery applied to the form $t^2-A{u}^2$ at the value $1$. (source: additions-7, Art. 75)

---

Historical Attribution (Add. VIII, Art. 85)

Lagrange traces the method:

• Fermat posed the problem $p^2=A{q}^2+1$ as a challenge to English mathematicians.
• Brouncker found the iterative solution algorithm.
• Wallis published it in his Algebra (1685, Ch. 98).
• Euler rediscovered and extended it; the misnomer “Pell’s equation” comes from Euler.
• Lagrange gave the first rigorous existence proof (Mélanges de Turin vol. IV; cleaner version in Add. II, Art. 37).

Lagrange also exhibits a counterexample in add8-pell-method-critique showing that the freedom of approximation direction (claimed by Wallis and Euler) can prevent the algorithm from terminating: for $p^2-6q^2=1$, taking the first limit “in minus” and all subsequent limits “in plus” produces a sequence whose leading coefficient never reaches $1$. See wallis-brouncker-method.

---

Related pages

• ch2.0.7-pell-equation-method
• ch2.0.6-integer-solutions-quadratic-squares
• ch2.0.11-quadratic-form-factorization
• ch2.0.12-quadratic-form-as-power
• ch2.0.13-impossibility-biquadrate-sums
• brahmagupta-fibonacci-identity
• quadratic-residues
• indeterminate-analysis
• linear-diophantine-equations
• add2-arithmetic-problems
• binary-quadratic-forms
• periodicity-quadratic-irrationals
• square-root-continued-fractions
• continued-fractions
• convergents
• add5-rational-quadratic-surds
• add7-integer-quadratic-method
• add8-pell-method-critique
• wallis-brouncker-method
• lagrange-reduction-algorithm
• norm-forms
• composition-of-forms