square-root-continued-fractions

Continued Fractions for Square Roots

Summary: Lagrange (Additions II, Art. 41, Scholium) reproduces Euler’s table from the New Commentaries of Petersburg vol. XI, listing the eventually-periodic continued-fraction expansion of $\sqrt{N}$ for every non-square $N$ from 2 to 99. These expansions are useful for solving Pell-type equations $p^2-N{q}^2=\pm 1$ since their convergents are exactly the candidate solutions.

Sources: additions-2

Last updated: 2026-05-10

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Form of the Expansion

Every $\sqrt{N}$ (with $N$ a non-square positive integer) has a continued fraction of the form

$\sqrt{N}=[c_0;\overline{c_1,c_2,\dots ,c_{r-1},2c_0}]$

where $c_0=\lfloor \sqrt{N}\rfloor$, the period $r$ is finite, and the inner block $c_1,\dots ,c_{r-1}$ is a palindrome (reads the same forwards and backwards). The last partial quotient before the period closes is always $2c_0$.

This regularity follows from the periodicity theorem (periodicity-quadratic-irrationals) plus a symmetry argument on the conjugate root.

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Table for $N\le 50$

(Modern notation: $[c_0;\overline{\text{period}}]$. Period length is the number of overlined entries.)

$N$	$\sqrt{N}$	Period
2	$[1;\overline{2}]$	1
3	$[1;\overline{1,2}]$	2
5	$[2;\overline{4}]$	1
6	$[2;\overline{2,4}]$	2
7	$[2;\overline{1,1,1,4}]$	4
8	$[2;\overline{1,4}]$	2
10	$[3;\overline{6}]$	1
11	$[3;\overline{3,6}]$	2
12	$[3;\overline{2,6}]$	2
13	$[3;\overline{1,1,1,1,6}]$	5
14	$[3;\overline{1,2,1,6}]$	4
15	$[3;\overline{1,6}]$	2
17	$[4;\overline{8}]$	1
18	$[4;\overline{4,8}]$	2
19	$[4;\overline{2,1,3,1,2,8}]$	6
20	$[4;\overline{2,8}]$	2
21	$[4;\overline{1,1,2,1,1,8}]$	6
22	$[4;\overline{1,2,4,2,1,8}]$	6
23	$[4;\overline{1,3,1,8}]$	4
24	$[4;\overline{1,8}]$	2
26	$[5;\overline{10}]$	1
27	$[5;\overline{5,10}]$	2
28	$[5;\overline{3,2,3,10}]$	4
29	$[5;\overline{2,1,1,2,10}]$	5
30	$[5;\overline{2,10}]$	2
31	$[5;\overline{1,1,3,5,3,1,1,10}]$	8
32	$[5;\overline{1,1,1,10}]$	4
33	$[5;\overline{1,2,1,10}]$	4
34	$[5;\overline{1,4,1,10}]$	4
35	$[5;\overline{1,10}]$	2
37	$[6;\overline{12}]$	1
38	$[6;\overline{6,12}]$	2
39	$[6;\overline{4,12}]$	2
40	$[6;\overline{3,12}]$	2
41	$[6;\overline{2,2,12}]$	3
42	$[6;\overline{2,12}]$	2
43	$[6;\overline{1,1,3,1,5,1,3,1,1,12}]$	10
44	$[6;\overline{1,1,1,2,1,1,1,12}]$	8
45	$[6;\overline{1,2,2,2,1,12}]$	6
46	$[6;\overline{1,3,1,1,2,6,2,1,1,3,1,12}]$	12
47	$[6;\overline{1,5,1,12}]$	4
48	$[6;\overline{1,12}]$	2
50	$[7;\overline{14}]$	1

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Notable Entries from the Full Table ($51\le N\le 99$)

$N$	$\sqrt{N}$	Period
53	$[7;\overline{3,1,1,3,14}]$	5
58	$[7;\overline{1,1,1,1,1,1,14}]$	7
61	$[7;\overline{1,4,3,1,2,2,1,3,4,1,14}]$	11
67	$[8;\overline{5,2,1,1,7,1,1,2,5,16}]$	10
73	$[8;\overline{1,1,5,5,1,1,16}]$	7
76	$[8;\overline{1,2,1,1,5,4,5,1,1,2,1,16}]$	12
79	$[8;\overline{1,7,1,16}]$	4
89	$[9;\overline{2,3,3,2,18}]$	5
94	$[9;\overline{1,2,3,1,1,5,1,8,1,5,1,1,3,2,1,18}]$	16
97	$[9;\overline{1,5,1,1,1,1,1,1,5,1,18}]$	11

The original table runs to $N=99$. $\sqrt{61}$ and $\sqrt{94}$ are notable for unusually long periods relative to nearby $N$.

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Lagrange’s Original Layout (Art. 41)

Lagrange formats each entry as a two-row block (per his $P^k,{\mu }^k$ algorithm of Art. 33 — see binary-quadratic-forms):

• Upper row: $P^0,-P^{\prime },P^{\prime \prime },-P^{\prime \prime \prime },\dots$ — the $|P^k|$ values with sign flipped to alternate
• Lower row: $\mu ,{\mu }^{\prime },{\mu }^{\prime \prime },{\mu }^{\prime \prime \prime },\dots$ — the partial quotients of $\sqrt{N}$

The lower row is the modern continued fraction. The upper row encodes the auxiliary data needed to detect the period and to compute $p^2-N{q}^2$ at each convergent (since $P^k=p^{k2}-N{q}^{k2}$ for $\sqrt{N}$).

Sample reading for $\sqrt{2}$:

P:  1 1 1 1 ...     →  P^k = 1, -1, 1, -1, ... (period 1, ±1 alternating)
μ:  1 2 2 2 ...     →  √2 = [1; 2̄]

So the convergents $p^k/q^k$ of $\sqrt{2}$ satisfy $p^{k2}-2q^{k2}=(-1{)}^k$, alternating $+1,-1,+1,\dots$ — exactly the Pell solutions for $K=2$.

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Period Length and Pell Solutions

For each non-square $N$:

• If the period $r$ is even, then $p^2-N{q}^2=-1$ is never solvable; the minimal Pell solution to $p^2-N{q}^2=+1$ is given by the convergent at index $r$.
• If the period $r$ is odd, then both $p^2-N{q}^2=-1$ and $=+1$ are solvable; the minimal $-1$-solution is at index $r$, and the minimal $+1$-solution is at index $2r$.

From the table:

$N$	Period	Solvable: $-1$?
2	1	yes
3	2	no
5	1	yes
13	5	yes
19	6	no
29	5	yes
41	3	yes
61	11	yes

This gives a quick test for which $N$ admit $p^2-N{q}^2=-1$: precisely those whose $\sqrt{N}$ has odd CF period.

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Convergents and Pell Solutions Compared with ch2.0.7-pell-equation-method

The table on pell-equation lists Euler’s minimal Pell solutions. These match exactly the convergents at the period boundary in Lagrange’s CF table:

$N$	Lagrange CF index $r$	Pell minimum $(p,q)$
2	1	(3, 2) at index 2 (since period odd, need 2r) — actually $(p_1,q_1)=(3,2)$, $p^2-2q^2=+1$ ✓
3	2	(2, 1): convergents $1/1,2/1$, last is the answer
5	1	(9, 4) at index 2
7	4	(8, 3): convergents $2/1,3/1,5/2,8/3$ ✓
13	5 (odd)	(649, 180) at index 10
29	5 (odd)	(9801, 1820) at index 10
61	11 (odd)	(1766319049, 226153980) at index 22 — explains the famously huge value

The huge minimal Pell solution at $N=61$ is forced by its CF having period 11 (odd) with non-trivial coefficients — running the convergent recurrence 22 times yields the giant $p,q$.

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Reference

Euler’s original table appeared in Novi Commentarii Academiae Scientiarum Petropolitanae, vol. XI (1765), supplemented in his Mémoires de l’Académie de Berlin 1768. Lagrange reproduces it as a service to his readers — anticipating modern reference tables.

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Related pages

• add2-arithmetic-problems — chapter context (Arts. 23–41)
• binary-quadratic-forms — the $P^k,Q^k,{\mu }^k$ algorithm
• periodicity-quadratic-irrationals — why the CF must be periodic
• pell-equation — Pell solutions read off from the table
• continued-fractions — definition and basics
• convergents — the $p^k/q^k$ that solve Pell
• ch2.0.7-pell-equation-method — Euler’s iterative descent (alternative algorithm)