Approximations of n-th root of integers ($2 \leq n \leq 5$) using algebraic methods

I present a way to find fractions that are as close as desired to a real number of the form $\sqrt[n]{a}$ where $a$ and $n$ are integers with $a$ greater than 1 and $2 \leq n \leq 5$. The main idea stems from the following observation: since $\lvert\sqrt{2} - 1\rvert < 1$ the sequence $u_m = (\sqrt{2} - 1)^m$ goes to zero as $m \to \infty$, since $u_m$ is of the form $a_m - b_m\sqrt{2}$, we derive a sequence of rational numbers $r_m = a_m/b_m$ which converges to $\sqrt{2}$. Of course this is already known and one can find the sequences $(a_m)$ and $(b_m)$ on the OEIS (Online Encyclopedia of Integer Sequences), $(a_m)$ is A001333, $(b_m)$ is A000129. Another known example is the approximation of $\sqrt{3}$ (see A026150 and A002605).
Below we will see that this can be generalized to cube, 4-th and 5-th roots with a slight adptation. Expressions approximating $\sqrt[3]{2}$ are derived, some are already known such as $63/50$ and $635/504$, some are less current such as $97/77$ and $790/627$, a byproduct of this method is the ability to find expressions approximating n-th root of integers while containing radicals of order at most $n-1$.

Approximation of square roots

Let $a$ be an integer $\geq 2$, we suppose that a is not a perfect square, to construct a sequence converging to $\sqrt{a}$, we start by finding $\left\lfloor \sqrt{a} \right\rfloor$ i.e. the integer square root of a, we denote it by r (there are several ways to find the integer square root of a number, one can use Heron's method for example). Since $\lvert\sqrt{a} - r\rvert \lt 1$, the sequence $(\sqrt{a}-r)^m$ converges to 0 as $m \to \infty$, also $(\sqrt{a}-r)^m = u_m - v_m\sqrt{a}$ where $(u_m)$ and $(v_m)$ are integer sequences (can be seen by using the binomial theorem and noticing that $\sqrt{a}^{2k} = a^{k}$ and $\sqrt{a}^{2k+1} = a^{k}\sqrt{a}$), it follows that the sequence $(u_m/v_m)$ converges to $\sqrt{a}$ as $m \to \infty$. The convergence can be sped up by taking the square of the previous iteration result, which is the same as using the sequence $(\sqrt{a}-d)^{2^{n}}$.

Here are a few examples:

a	n	$(\sqrt{a}-1)^n$	approximant	precision
2	2	$3-2\sqrt{2}$	$3/2$	$8.57864\cdot10^{-2}$
2	3	$-7+5\sqrt{2}$	$7/5$	$1.42136\cdot10^{-2}$
2	4	$17-12\sqrt{2}$	$17/12$	$2.45310\cdot10^{-3}$
2	5	$41+29\sqrt{2}$	$41/29$	$4.20459\cdot10^{-4}$
2	6	$99-70\sqrt{2}$	$99/70$	$7.21519\cdot10^{-5}$
2	7	$-239+169\sqrt{2}$	$239/169$	$1.23789\cdot10^{-5}$
2	8	$577-408\sqrt{2}$	$577/408$	$2.12390\cdot10^{-6}$
2	9	$-1393+985\sqrt{2}$	$1393/985$	$3.64404\cdot10^{-7}$
2	10	$3363-2378\sqrt{2}$	$3363/2378$	$6.25218\cdot10^{-8}$
2	16	$665857 - 470832\sqrt{2}$	$665857/470832$	$1.59486\cdot10^{-12}$
2	32	$886731088897 - 627013566048\sqrt{2}$	$886731088897/627013566048$	$8.99293\cdot10^{-25}$
3	2	$4 - 2\sqrt{3}$	$2$	$2.67949\cdot10^{-1}$
3	3	$-10 + 6\sqrt{3}$	$5/3$	$6.53841\cdot10^{-2}$
3	4	$28 - 16\sqrt{3}$	$7/4$	$1.79492\cdot10^{-2}$
3	5	$-76 + 44\sqrt{3}$	$19/11$	$4.77808\cdot10^{-3}$
3	6	$208 - 120\sqrt{3}$	$26/15$	$1.28253\cdot10^{-3}$
3	7	$-568 + 328\sqrt{3}$	$71/41$	$3.43490\cdot10^{-4}$
3	8	$1552 - 896\sqrt{3}$	$97/56$	$9.20496\cdot10^{-5}$
3	9	$-4240 + 2448\sqrt{3}$	$265/153$	$2.46638\cdot10^{-5}$
3	10	$11584 - 6688\sqrt{3}$	$362/209$	$6.60870\cdot10^{-6}$
3	16	$4817152 - 2781184\sqrt{3}$	$18817/10864$	$2.44585\cdot10^{-9}$
3	32	$46409906716672 - 26794772135936\sqrt{3}$	$708158977/408855776$	$1.72691\cdot10^{-18}$

Note that the sequence of numerators for the approximation of $\sqrt{3}$ is A002531 and the sequence of denominators is A002530.

Approximation of $\sqrt[3]{2}$

If we try to apply the same method for approximating $\sqrt[3]{2} = 2^{\frac{1}{3}}$, we get $(\sqrt[3]{2}-1)^{2} = 1+2^{2/3}-2\cdot2^{1/3}$ from which we can not derive an approximation as we did in the case of square roots because in addition to $2^{1/3}$ there is the term $2^{2/3}$. However if we look at the expression $1+2^{2/3}-2\cdot2^{1/3}$ as being the evaluation at $X=2^{1/3}$ of the polynomial $P(X) = X^{2}-2X+1$, then if $P$ has real roots one of them has to be close to $2^{1/3}$. In this case P has $1$ as unique real root (with multiplicity 2), so the approximation we get is $2^{1/3} \approx 1$, which is not really interesting. Let us see what we get if we move to exponent 3: $(2^{1/3}-1)^{3} = (1+2^{2/3}-2\cdot2^{1/3})(2^{1/3}-1) = 1 + 3\cdot2^{1/3} - 3\cdot2^{2/3}$ which is the value of $P(X) = -3X^{2}+3X+1$ at $X=\sqrt[3]{2}$, now $P$ has the roots $(3 \pm \sqrt{21})/6$, which produces the approximation

with precision $3.84e-3$. This approximant of $\sqrt[3]{2}$ contains a square root on which we can apply the method described in the previous section: we have $(\sqrt{21}-5)^{2} = 46-10\sqrt{21}$ hence $\sqrt{21} \approx 23/5$ from which we deduce the approximation

with precision $6.7456e-3$.
With exponent 4: $(2^{1/3}-1)^{4} = (1 + 3\cdot2^{1/3} - 3\cdot2^{2/3})(2^{1/3}-1) = -7 - 2\cdot2^{1/3} + 6\cdot2^{2/3}$, the polynomial is now $P(X) = 6X^{2}-2X-7$, the roots of P are $(1 \pm \sqrt{43})/6$. The positive root yields

the precision being $3.47963e-4$. By approximating $\sqrt{43}$ we get

where each approximation is followed by its precision between parenthesis. Below is a table giving the approximations that can be obtained with next few exponents

n	Polynomial	Approximant	Precision
$5$	$-8X^{2}-5X+19$	$\frac{\sqrt{633}-5}{16}$	$4.71533e-5$
		$\frac{5}{4}$	$9.92105e-3$
		$\frac{63}{50}$	$7.89501e-5$
		$\frac{790}{627}$	$4.70522e-5$
		$\frac{19813}{15725}$	$4.71536e-5$
$6$	$3X^{2}+24X-35$	$\sqrt{\frac{83}{3}}-4$	$9.77054e-6$
		$1$	$2.59921e-1$
		$\frac{19}{15}$	$6.74562e-3$
		$\frac{97}{77}$	$1.80790e-4$
		$\frac{1493}{1185}$	$5.43808e-6$
		$\frac{7659}{6079}$	$9.88029e-6$
		$\frac{117871}{93555}$	$9.76776e-6$

Approximation of n-th roots ($2 \leq n \leq 5$)

General principle

The method described for the approximation of $\sqrt[3]{2}$ in the previous section can be generalized as follows. Let a be a positive integer such that there exists no integer b such that $b^{n} = a$, we wish to approximate $\sqrt[n]{a}$. Suppose we have computed the integer n-th root of a, i.e. $\lfloor \sqrt[n]{a} \rfloor$, we call it d. The minimal polynomial of $\sqrt[n]{a}$ over $\mathbb{Q}$ is $X^{n} - a$, hence the algebraic extension $\mathbb{Q}(\sqrt[n]{a})$ of $\mathbb{Q}$ is of degree n, it is a $\mathbb{Q}$-vector space of dimension n, of which a basis is $(1, \sqrt[n]{a}, (\sqrt[n]{a})^{2}, \dots, (\sqrt[n]{a})^{n-1})$. Hence for each $m \in \mathbb{N}$, $(\sqrt[n]{a}-d)^m$ can be written as $a_0 + a_1\cdot\sqrt[n]{a} + a_2\cdot(\sqrt[n]{a})^2 + \dots + a_{n-1}\cdot(\sqrt[n]{a})^{n-1}$ which is the value taken by the polynomial $P_m(X) = a_0 + a_1X + \dots + a_{n-1}X^{n-1}$ at $\sqrt[n]{a}$, since $(\sqrt[n]{a}-d)^m$ converges to $0$ as $m$ goes to infinity, we see that $\sqrt[n]{a}$ is getting closer and closer to one of the real roots of $P_m$ if such roots exist. Since the degree of $P_m$ is at most $n-1$, it ensues that for $n \leq 5$ finding the roots of $P_m$ is done by solving an algebraic equation of degree less than or equal to 4, which we know can be done in radicals. Thus finding the roots of $P_m$ yields an approximation of $\sqrt[n]{a}$ by an expression containing radicals of order at most n-1, by repeating this whole process for the (possibly) nested radicals one derives a fraction approximating $\sqrt[n]{a}$. Several points can be noted:

• computing the polynomial $P_m$ is equivalent to reducing $(X - d)^m$ in $\mathbb{Q}[X]/(X^{n}-a)$
• the whole process can be done with d = $\lceil \sqrt[n]{a} \rceil$ since we only need $\lvert\sqrt[n]{a}-d\rvert \lt 1$. Actually $a$ does not need to be an integer, it can be any rational, also $d$ does not have to be either $\lfloor \sqrt[n]{a} \rfloor$ or $\lceil \sqrt[n]{a} \rceil$, in fact $d$ can be any rational number $q$ such that $\lvert\sqrt[n]{a}-q\rvert \lt 1$.
• Regarding the polynomials since we are actually working in $\mathbb{Q}[X]/(X^{n}-a)$, $P_m$ will not produce a useful approximation until m >= n.

Additional examples

Let us treat the case of $\sqrt[5]{2}$, here obviously $d = 1$, the polynomial is $(X-1)^{5}\ mod\ (X^{5}-2) = -5\cdot X^{4} + 10\cdot X^{3} - 10\cdot X^{2} + 5\cdot X + 1$, it has two non-complex roots:

The positive root yields

with a precision of 8.35319e-6. If we replace $\sqrt{5}$ by its integer part we get

If we replace $\sqrt{35}$ by its approximations 6 and $\frac{65}{11}$ we get

If we proceed with a more precise approximation of $\sqrt{5}$ such as $9/4$ we get

To get an approximation of $\sqrt{85/2}$, it is enough to compute $(\sqrt{85/2} - 6)^{2} = (157-24\sqrt{85/2})/2$ which gives us $\sqrt{85/2}\approx157/24\quad\textit{(2.24643e-2)}$ from which we derive

If we repeat this process with approximations of $\sqrt{5}$ that are more and more precise we can get closer and closer to the precision of (1). The appendices contains a table which reproduces these results and extends them.

Automated generation of the approximations

I wrote a PARI/GP script which can generate this kind of approximations. It is based on the principles described in this page. To invoke the script, start by installing PARI/GP, run it and then call function approximate_nthroot(n, a, depth):

$ gp
gp > \r proximar.gp
gp > approximate_nthroot(2, 5, 1);

which will output approximations of $\sqrt{5}$. Here is what the output looks like:

| 2 | 2.36067977 e-1 |
| 9/4 | 1.39320225 e-2 |
| 38/17 | 7.73859853 e-4 |
| 161/72 | 4.31336113 e-5 |
| 682/305 | 2.40372930 e-6 |
| 2889/1292 | 1.33955319 e-7 |
| 12238/5473 | 7.46507382 e-9 |
| 51841/23184 | 4.16014306 e-10 |
| 219602/98209 | 2.31836827 e-11 |

Here is an example for a cube root, $\sqrt[3]{17}$:

gp > approximate_nthroot(3, 17, 4);
| 3 | 4.28718409 e-1 |
| (9+sqrt(41))/6 | 4.09421775 e-3 |
| 5/2 | 7.12815907 e-2 |
| 185/72 | 1.83714621 e-3 |
| 765/298 | 4.16749670 e-3 |
| 28465/11088 | 4.09183597 e-3 |
| 117685/45842 | 4.09429517 e-3 |
| (91+sqrt(34849))/108 | 1.80955021 e-4 |
| 139/54 | 2.79248342 e-3 |
| 25963/10098 | 1.78401908 e-4 |
| 4850911/1886706 | 1.80952827 e-4 |
| 906341467/352511082 | 1.80955019 e-4 |
| 169340326999/65862971154 | 1.80955021 e-4 |
| (75+6*sqrt(9201))/253 | 1.25823193 e-5 |
| 651/253 | 1.84093899 e-3 |
| 20817/8096 | 1.18278124 e-5 |
| 7990473/3107599 | 1.25820120 e-5 |
| 255591051/99402688 | 1.25823191 e-5 |
| 98107011219/38155094197 | 1.25823193 e-5 |
| (-279+sqrt(3375289))/606 | 1.12656428 e-6 |
| 779/303 | 3.24494949 e-4 |
| 1431203/556611 | 1.10931669 e-6 |
| 2629260131/1022548947 | 1.12656520 e-6 |
| 4830208477187/1878522605619 | 1.12656428 e-6 |
| 8873566239416099/3451030085332563 | 1.12656428 e-6 |

And for $\sqrt[4]{19}$:

| 2 | 8.77976299 e-2 |
| 1 + ((19/8 + 1/24*sqrt(10003/3))/2)^(1/3) - ((-19/8 + 1/24*sqrt(10003/3))/2)^(1/3) | 1.63244203 e-6 |
| 1 + 1/2*(115/6)^(1/3) - 1/2*(1/6)^(1/3) | 2.48675893 e-2 |
| 9/4 | 1.62202370 e-1 |
| 1 + 1/2*((27+sqrt(447))/18) - 1/2*((3/4+1/4*sqrt(13))/3) | 2.57407522 e-2 |
| 49/24 | 4.61309633 e-2 |
| 8311/4032 | 2.65377093 e-2 |
| 890543/431880 | 2.57827184 e-2 |
| 17842283/8652672 | 2.57429259 e-2 |
| 5083361733/2465190008 | 2.57408649 e-2 |
| 1 + 1/2*((533/6+1/6*sqrt(1442713))/108) - 1/2*((1/3+5/6*sqrt(5/2))/3) | 2.48926181 e-2 |
| 433/216 | 8.31680003 e-2 |
| 711427/345888 | 3.09844418 e-2 |
| 2070378533/1003965384 | 2.55965162 e-2 |
| 2973454099175/1441449116224 | 2.49748319 e-2 |
| 5116770554669967/2480382941881048 | 2.49022325 e-2 |
| 1 + 1/2*((235/2+1/2*sqrt(6002985))/1505/3) - 1/2*((-5/24+1/8*sqrt(1291/3))/13/3) | 2.48691669 e-2 |
| 64461/31304 | 2.86039167 e-2 |
| 474642499/230084400 | 2.48915876 e-2 |
| 135931278126781/65892390167240 | 2.48693042 e-2 |
| 2979110623495558171/1444117272201895200 | 2.48691678 e-2 |
| 2559409631914250914578981/1240668146204832707010376 | 2.48691670 e-2 |
| 1 + 1/2*((-2259/2+1/2*sqrt(401741203/3))/1740) - 1/2*((-7/2+1/3*sqrt(973/2))/7) | 2.48676801 e-2 |
| 100571/48720 | 2.35324411 e-2 |
| 6978323029/3382727040 | 2.48694225 e-2 |
| 626163467661733469/303531133975350960 | 2.48676779 e-2 |
| 2418447663789413883467/1172336299642154169600 | 2.48676801 e-2 |
| 1302037558106569125953006574937/631159365455068772146904413040 | 2.48676801 e-2 |
| 1 + 1/4*(39931/261)^(1/3) - 1/4*(259/261)^(1/3) | 5.94350537 e-5 |
| 2 | 8.77976299 e-2 |
| 1 + 1/4*((75+sqrt(635495/87))/30) - 1/4*((3+5*sqrt(31/87))/6) | 6.42708859 e-5 |
| 101/48 | 1.63690367 e-2 |
| 164537/78880 | 1.88231553 e-3 |
| 324286513/155307152 | 2.35720910 e-4 |
| 26355444619547/12623265946560 | 4.91072871 e-5 |
| 257270583322774267/123221951666650128 | 6.56106944 e-5 |
<... truncated ...>

Note that the case n = 5, is not yet implemented.

Appendices

Approximation of $\sqrt[3]{2}$

n	Polynomial	Approximations	Precision
3	$-3X^{2} + 3X + 1$	(3+sqrt(21))/6	3.841566 e-3
		4/3	7.341228 e-2
		19/15	6.745617 e-3
		91/72	3.967839 e-3
		436/345	3.847066 e-3
		2089/1653	3.841806 e-3
		10009/7920	3.841576 e-3
4	6X^2 - 2X - 7	(1+sqrt(43))/6	3.479625 e-4
		4/3	7.341228 e-2
		53/42	1.983712 e-3
		359/285	2.719271 e-4
		4867/3864	3.454805 e-4
		32992/26193	3.478815 e-4
		447287/355110	3.479599 e-4
5	$-8X^{2} - 5X + 19$	(-5+sqrt(633))/16	4.715328 e-5
		5/4	9.921050 e-3
		63/50	7.895011 e-5
		790/627	4.705218 e-5
		19813/15725	4.715360 e-5
		496905/394379	4.715328 e-5
		12462251/9890925	4.715328 e-5
6	$3X^{2} + 24*X - 35$	(-12+sqrt(249))/3	9.770542 e-6
		4/3	7.341228 e-2
		121/96	4.956168 e-4
		3844/3051	6.267856 e-6
		122161/96960	9.746265 e-6
		3882244/3081363	9.770373 e-6
		123376681/97924896	9.770541 e-6
7	$21X^{2} - 59X + 41$	(59-sqrt(37))/42	1.317557 e-5
		53/42	1.983712 e-3
		635/504	4.149742 e-7
		7673/6090	1.326866 e-5
		92711/73584	1.317494 e-5
		1120205/889098	1.317558 e-5
		13535171/10742760	1.317557 e-5
8	$-80X^{2} + 100X + 1$	(25+sqrt(645))/40	2.050651 e-7
		5/4	9.921050 e-3
		63/50	7.895011 e-5
		635/504	4.149742 e-7
		16001/12700	2.099476 e-7
		20160/16001	2.050267 e-7
		2032001/1612800	2.050654 e-7
9	$180X^{2} - 99X - 161$	(33+sqrt(13969))/120	1.527106 e-8
		151/120	1.587717 e-3
		35681/28320	1.266489 e-6
		8427511/6688920	1.630500 e-8
		1990498241/1579859520	1.527023 e-8
		470136822871/373147848120	1.527106 e-8
		111041862618401/88133985834720	1.527106 e-8
10	$-279X^{2} - 62X + 521$	(-31+4*sqrt(9145))/279	1.839639 e-9
		353/279	5.311925 e-3
		16873/13392	1.025238 e-5
		12933401/10265247	2.165850 e-8
		619605265/491781024	1.877958 e-9
		474938572049/376958993895	1.839713 e-9
		22753059484537/18059115254256	1.839639 e-9

Approximation of $\sqrt[3]{3}$

n	Polynomial	Approximations	Precision
1	$X - 1$	1	4.42250e-1
3	$-3X^{2}+3X+2$	$(1 + \sqrt{11/3})/2$	1.51775e-2
		$3/2$ ($\sqrt{11/3} \approx 2$)	5.77504e-2
		$4/3$ ($\sqrt{11/3} \approx 5/3$)	1.08916e-1
		$22/15$ ($\sqrt{11/3} \approx 29/15$)	2.44171e-2
		$118/81$ ($\sqrt{11/3} \approx 115/81$)	1.45406e-2
		$634/435$ ($\sqrt{11/3} \approx 833/435$)	1.52217e-2
4	$6X^{2}-X-11$	$(1 + \sqrt{265})/12$	2.34785e-3
		$17/12$ ($\sqrt{265} \approx 16$)	2.55829e-2
		$553/384$ ($\sqrt{265} \approx 521/32$)	2.14540e-3
		$17849/12396$ ($\sqrt{265} \approx 16816/1033$)	2.34960e-3
5	$-7X^{2}-10X+29$	$(2\sqrt{57} - 5)/7$	5.60268e-4
		$9/7$ ($\sqrt{57} \approx 7$)	1.56535e-1
		$71/49$ ($\sqrt{57} \approx 53/7$)	6.73002e-3
		$515/357$ ($\sqrt{57} \approx 385/51$)	3.27461e-4
		$3747/2597$ ($\sqrt{57} \approx 2801/371$)	5.69067e-4
6	$-3X^{2}+39X-50$	$(13 - \sqrt{307/3})/2$	2.46539e-4
		$3/2$ ($\sqrt{307/3} \approx 10$)	5.77504e-2
		$173/120$ ($\sqrt{307/3} \approx 607/60$)	5.82904e-4
		$3481/2414$ ($\sqrt{307/3} \approx 12210/1207$)	2.44599e-4
		$210071/145680$ ($\sqrt{307/3} \approx 736849/72840$)	2.46550e-4
7	$42X^{2}-89X+41$	$(89 + \sqrt{1033})/84$	1.02935e-4
		$121/84$ ($\sqrt{1033} \approx 32$)	1.77338e-3
		$7753/5376$ ($\sqrt{1033} \approx 2057/64$)	9.92727e-5
		$497281/344820$ ($\sqrt{1033} \approx 131936/4105$)	1.02943e-4
8	$-131X^{2}+130X+85$	$(65 + 32\sqrt{15})/131$	5.90358e-6
		$193/131$ ($\sqrt{15} \approx 4$)	3.10329e-2
		$1319/917$ ($\sqrt{15} \approx 27/7$)	3.86353e-3
		$189/131$ ($\sqrt{15} \approx 31/8$)	4.98521e-4
		$10391/7205$ ($\sqrt{15} \approx 213/55$)	5.66487e-5
		$5857/4061$ ($\sqrt{15} \approx 1921/496$)	6.03176e-6
		$644081/446579$ ($\sqrt{15} \approx 13203/3409$)	5.88729e-6
		$737793/511555$ ($\sqrt{15} \approx 15124/3905$)	5.90564e-6
9	$261X^{2}-45X-478$	$5/58 + \sqrt{55673}/174$	9.14246e-7
		$251/174$ ($\sqrt{55673}\approx236$)	2.79165e-4
		$9829/6815$ ($\sqrt{55673}\approx55449/235$)	1.01509e-5
		$6943445/4814319$ ($\sqrt{55673}\approx13056835/55337$)	9.36596e-7
		$1635012119/1133654805$ ($\sqrt{55673}\approx3074566513/13030515$)	9.14201e-7
		$385005513805/266948082903$ ($\sqrt{55673}\approx723985496075/3068368769$)	9.14246e-7
10	$-306X^{2}-433X+1261$	$(\sqrt{1730953} - 433)/612$	2.17536e-7
		$49/34$ ($\sqrt{1730953}\approx1315$)	1.07310e-3
		$883/612$ ($\sqrt{1730953}\approx1316$)	5.60887e-4
		$64483/44710$ ($\sqrt{1730953}\approx1730089/1315$)	4.85608e-7
		$331879/230112$ ($\sqrt{1730953}\approx494687/376$)	2.90630e-7
		$84816313/58808338$ ($\sqrt{1730953}\approx2275635115/1729657$)	2.17469e-7
		$873391621/605575836$ ($\sqrt{1730953}\approx1301846420/989503$)	2.17546e-7
		$111561308251/77352279190$ ($\sqrt{1730953}\approx2993207574673/2275067035$)	2.17536e-7
		$328352437105/227666829888$ ($\sqrt{1730953}\approx489430439297/372004624$)	2.17536e-7
		$146739760997281/101743652336866$ ($\sqrt{1730953}\approx3937051035064675/2992460362849$)	2.17536e-7
		$864110946941251/599140976982372$ ($\sqrt{1730953}\approx1288012978041524/978988524481$)	2.17536e-7

Between parenthesis are the approximations used for the square root appearing in the root of the polynomial.

Approximation of $\sqrt{5}$

Exponent of $\sqrt{5}-2$	Approximant	Precision
1	2	2.36068e-1
2	9/4	1.39320e-2
3	38/17	7.73860e-4
4	161/72	4.31336e-5
5	682/305	2.40373e-6
6	2889/1292	1.33955e-7
7	12238/5473	7.46507e-9
8	51841/23184	4.16014e-10
9	219602/98209	2.31837e-11
10	930249/416020	1.29198e-12

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Authored by Youcef Lemsafer. Creation date: 2025.02.20.

This file is licensed under CC BY 4.0. The script is licensed under the MIT license.