periodicity-quadratic-irrationals

Periodicity of Continued Fractions for Quadratic Irrationals

Summary: Lagrange’s theorem (Additions II, Art. 34) states that the continued-fraction expansion of any quadratic irrational is eventually periodic. The proof shows that the auxiliary $P^k,Q^k$ sequences appearing in the minimization of a binary quadratic form are bounded (by $E$ and $\sqrt{E}$ respectively), forcing recurrence on a finite integer lattice.

Sources: additions-2

Last updated: 2026-05-10

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Statement

Let $\alpha$ be a real irrational satisfying a monic quadratic with integer coefficients,

$A{\alpha }^2+B\alpha +C=0(A,B,C\in \mathbb{Z},B^2-4AC\text{ positive non-square}).$

Then the continued-fraction expansion $\alpha =[\mu ;{\mu }^{\prime },{\mu }^{\prime \prime },\dots ]$ is eventually periodic: there exist $\pi \ge 0$ and $\rho \ge 1$ such that

${\mu }^{k+2\rho }={\mu }^k\text{for all }k\ge \pi .$

A CF that is purely periodic (i.e. $\pi =0$) corresponds to a “reduced” surd; otherwise there is a finite pre-period followed by the repeating block.

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Lagrange’s Proof (Add. II, Arts. 33–34)

Setup. Run the binary-quadratic-form algorithm of Add. II, Art. 33 (see binary-quadratic-forms) on $F(x,1)=A{x}^2+Bx+C$ with root $\alpha$. This generates auxiliary integer sequences

$P^0=A,Q^0=\frac{1}{2}B,$ $P^{k+1}={\mu }^{k2}{P}^k+2{\mu }^k{Q}^k+P^{k-1},Q^{k+1}={\mu }^k{P}^k+Q^k$

with the fundamental identity

$Q^{k2}-P^{k-1}{P}^k=\frac{1}{4}E,E=B^2-4AC.$

Step 1 — Eventual sign alternation (Art. 34, ¶1). After finitely many steps, two consecutive $P^{\pi },P^{\pi +1}$ have opposite signs. From that point on, all $P^k$ alternate in sign.

Reason: The principal convergents $p^k/q^k$ alternate around $\alpha$, hence $p^k-\alpha q^k$ alternates in sign; multiplying by $A$ and accounting for the conjugate factor $p^k-\beta q^k$ (where $\beta$ is the other root) shows $P^k=A(p^k-\alpha q^k)(p^k-\beta q^k)$ alternates in sign once both factors stabilize.

Step 2 — Bound on $|P^k|$ (Art. 34, ¶2). Once the sign-alternating regime starts (at index $\lambda$), the products $P^k{P}^{k+1}$ are negative; by the fundamental identity,

$Q^{k+12}=P^k{P}^{k+1}+\frac{1}{4}E\le \frac{1}{4}E,$

so $|Q^{k+1}|\le \frac{1}{2}\sqrt{E}$. Consequently $|P^k|\cdot |P^{k+1}|\le \frac{1}{4}E$, and since both are positive integers,

$|P^k|\le E,|Q^k|\le \sqrt{E}.$

Step 3 — Pigeonhole (Art. 34, ¶3). The integer pair $(P^k,Q^k)$ takes values in a finite set of size $\le 2E\cdot 2\sqrt{E}=4E^{3/2}$. By the pigeonhole principle, some pair recurs; let $\pi$ be the first index where the pair $(P^{\pi },Q^{\pi })$ recurs at index $\pi +2\rho$. Then by the deterministic recurrence,

$P^{\pi +2\rho +j}=P^{\pi +j},Q^{\pi +2\rho +j}=Q^{\pi +j},{\mu }^{\pi +2\rho +j}={\mu }^{\pi +j}$

for all $j\ge 0$. The CF is therefore eventually periodic with period $2\rho$. ∎

The factor of 2 in the period (Lagrange takes $2\rho$ rather than $\rho$) ensures the sign pattern of $P^k$ also matches at the boundary; the true period of $({\mu }^k)$ may be either $\rho$ or $2\rho$ depending on the surd.

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Consequence: Pell Equation Always Solvable (Add. II, Art. 37)

Specializing to $A=1,B=0,C=-K$ ($K$ non-square positive), the form is $p^2-K{q}^2$ with $E=4K,\sqrt{E}/2=\sqrt{K}$. Then $P^0=1$, and after one full period $P^{2\rho }=P^0=1$, giving

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

This is the Pell solution. See pell-equation for the broader context.

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Lagrange’s “Reverse Period” Observation

For the example $9p^2-118pq+378q^2$ (Add. II, Art. 40), the two roots

$a_1=\frac{59+\sqrt{79}}{9},a_2=\frac{59-\sqrt{79}}{9}$

have CF expansions

$a_1=[7;\overline{1,1,5,3,2,1}],a_2=[5;\overline{1,1,3,5,1,1}]$

(both period 6). The two periodic blocks are reverses of each other (read $1,1,5,3,2,1$ backwards: $1,2,3,5,1,1$ — matching $a_2$‘s block up to the boundary alignment). This is a general feature of conjugate quadratic surds, formalized later by Galois (1828) for purely periodic CFs.

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Modern Statement and Converse

Galois–Lagrange theorem (modern): A real irrational has an eventually periodic CF expansion if and only if it is a quadratic irrational.

The “if” direction is Lagrange’s theorem above; the “only if” direction is straightforward (a periodic CF satisfies a quadratic equation by direct substitution). Galois (1828) proved that the CF is purely periodic iff $\alpha >1$ and its conjugate $\beta$ satisfies $-1<\beta <0$ — these are the reduced quadratic surds.

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Length of the Period

The period length is bounded above by $4E^{3/2}$ via the pigeonhole bound, but in practice it is much smaller. For $\sqrt{N}$ with $N\le 100$ (Add. II, Art. 41 — see square-root-continued-fractions):

$N$ range	typical period length
2–20	1–4
21–60	4–10
61–99	5–18
Maximum in this range	18 (at $N=94$)

The periods of $\sqrt{N}$ exhibit a curious palindromic pattern: $[c_0;\overline{c_1,c_2,\dots ,c_{r-1},2c_0}]$ where $c_1,\dots ,c_{r-1}$ is itself a palindrome.

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Why the Theorem Was Needed

Before Lagrange:

• Pell-equation solvability had been established empirically by Fermat (in challenges to Wallis and Brouncker, ~1657) and worked out case-by-case by Brouncker, Wallis, and Euler.
• Existence in the general case was assumed but never proved.
• Euler’s iterative method (in ch2.0.7-pell-equation-method) finds the minimal solution but rests on the assumption that one exists.

Lagrange’s periodicity theorem is the missing existence proof, made rigorous by reducing the problem to a pigeonhole argument on a finite integer lattice.

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Related pages

• add2-arithmetic-problems — full chapter context (Arts. 23–41)
• binary-quadratic-forms — the $P^k,Q^k$ algorithm
• continued-fractions — CF basics
• convergents — convergent recurrences
• pell-equation — solvability via this theorem
• square-root-continued-fractions — Euler’s table illustrating periodicity
• infinite-descent — alternative proof technique (not used here)